A General Theory of Tensor Products of Convex Sets in Euclidean Spaces

TL;DR

This paper develops a unified geometric framework for tensor products of convex sets in Euclidean spaces, establishing a correspondence with tensor norms and extending classical theorems like Grothendieck's.

Contribution

It introduces tensor products of convex bodies, links them to tensor norms, and generalizes Grothendieck's theorem in a geometric setting.

Findings

Bijective correspondence between tensor products of convex bodies and tensor norms.

Formulation of Grothendieck's theorem for convex bodies.

Geometric representation of Hilbertian tensor product.

Abstract

We introduce both the notions of tensor product of convex bodies that contain zero in the interior, and of tensor product of $0$-symmetric convex bodies in Euclidean spaces. We prove that there is a bijection between tensor products of $0$-symmetric convex bodies and tensor norms on finite dimensional spaces. This bijection preserves duality, injectivity and projectivity. We obtain a formulation of Grothendieck`s Theorem for $0$-symmetric convex bodies and use it to give a geometric representation (up to the $K_G$-constant) of the Hilbertian tensor product. We see that the property of having enough symmetries is preserved by these tensor products, and exhibit relations with the L\"owner and the John ellipsoids.

Equations

$$T_1\otimes \cdots \otimes T_l\left(P_1{\otimes }_{\alpha }\cdots {\otimes }_{\alpha }{P}_l\right)\subseteq Q_1{\otimes }_{\alpha }\cdots {\otimes }_{\alpha }{Q}_l.$$

$$\otimes :V_1\times \cdots \times V_l$$

$$\left(x^1,\dots ,x^l\right)$$

$${⟨x^1\otimes \cdots \otimes x^l,y^1\otimes \cdots \otimes y^l⟩}_H:={\Pi }_{i=1}^l{⟨x^i,y^i⟩}_{{\mathbb{E}}_i},$$

$$g_P\left(x\right):=\inf \left\{\lambda >0:{\lambda }^{-1}x\in P\right\}\text{ for }x\in \mathbb{E}.$$

$$\mathcal{B}\left(\mathbb{E}\right)$$

$$P$$

$$\pi \left(u\right):=\inf \left\{{\sum }_{j=1}^n\Vert x_j^1\Vert \cdots \Vert x_j^l\Vert :u={\sum }_{j=1}^n{x}_j^1\otimes \cdots \otimes x_j^l\right\}$$

$$ϵ\left(u\right):=\sup \left\{\left|x_1^{\ast }\otimes \cdots \otimes x_l^{\ast }\left(u\right)\right|:x_i^{\ast }\in B_{X_i^{\ast }},\text{ for }i=1,\dots ,l\right\}.$$

$$ϵ\left(u\right)\le \alpha \left(u\right)\le \pi \left(u\right)\text{ for every }u\in {\otimes }_{i=1}^l{X}_i.$$

$$P_1{\otimes }_{\pi }\cdots {\otimes }_{\pi }{P}_l:=\text{conv}\left\{x^1\otimes \cdots \otimes x^l\in {\otimes }_{i=1}^l{\mathbb{E}}_i:x^i\in P_i,i=1,\dots ,l\right\}.$$

$$P_1{\otimes }_{ϵ}\cdots {\otimes }_{ϵ}{P}_l:={\left(P_1^{\circ }{\otimes }_{\pi }\cdots {\otimes }_{\pi }{P}_l^{\circ }\right)}^{\circ }.$$

$$T_1\otimes \cdots \otimes T_l\left(P_1{\otimes }_{\pi }\cdots {\otimes }_{\pi }{P}_l\right)\subseteq Q_1{\otimes }_{\pi }\cdots {\otimes }_{\pi }{Q}_l$$

$$T_1\otimes \cdots \otimes T_l\left(P_1{\otimes }_{ϵ}\cdots {\otimes }_{ϵ}{P}_l\right)\subseteq Q_1{\otimes }_{ϵ}\cdots {\otimes }_{ϵ}{Q}_l.$$

$$T_1\otimes \cdots \otimes T_l(P_1{\otimes }_{\pi }\cdots {\otimes }_{\pi }{P}_l)=\text{conv}\left\{T_1{x}^1\otimes \cdots \otimes T_l{x}^l\in {\otimes }_{i=1}^l{\mathbb{F}}_i:x^i\in P_i\right\}.$$

$$(P_1\cap M_1){\otimes }_{ϵ}\cdots {\otimes }_{ϵ}(P_l\cap M_l)=P_1{\otimes }_{ϵ}\cdots {\otimes }_{ϵ}{P}_l\cap {\otimes }_{i=1}^l{M}_l.$$

$$T_1\otimes \cdots \otimes T_l\left(P_1{\otimes }_{\pi }\cdots {\otimes }_{\pi }{P}_l\right)=T_{1}P{\otimes }_{\pi }\cdots {\otimes }_{\pi }{T}_{l}P.$$

$$P_1{\otimes }_{\pi }\cdots {\otimes }_{\pi }{P}_l=B_{{\otimes }_{\pi ,i=1}^l\left({\mathbb{E}}_i,g_{P_i}\left(\cdot \right)\right)}\text{mbox}and{P}_1{\otimes }_{ϵ}\cdots {\otimes }_{ϵ}{P}_l=B_{{\otimes }_{ϵ,i=1}^l\left({\mathbb{E}}_i,g_{P_i}\left(\cdot \right)\right)}.$$

$$Q_1{\otimes }_{\pi }\cdots {\otimes }_{\pi }{Q}_l\subseteq Q\subseteq Q_1{\otimes }_{ϵ}\cdots {\otimes }_{ϵ}{Q}_l.$$

$${\mathcal{E}}_1{\otimes }_2\cdots {\otimes }_2{\mathcal{E}}_l:=T_1\otimes \cdots \otimes T_l\left(B_2^{d_1,\dots ,d_l}\right).$$

$$Q^{\circ }=\left\{z\in {\otimes }_{i=1}^l{\mathbb{E}}_i:{\sup }_{u\in Q}{⟨u,z⟩}_H\le 1\right\}.$$

$$T_1\otimes \cdots \otimes T_l\left(P_1{\otimes }_{\alpha }\cdots {\otimes }_{\alpha }{P}_l\right)\subseteq Q_1{\otimes }_{\alpha }\cdots {\otimes }_{\alpha }{Q}_l.$$

$$T_1\otimes \cdots \otimes T_l\left(P_1{\otimes }_{\alpha }\cdots {\otimes }_{\alpha }{P}_l\right)=T_1{P}_1{\otimes }_{\alpha }\cdots {\otimes }_{\alpha }{T}_l{P}_l.$$

$$P_1{\otimes }_{{\alpha }^{\prime }}\cdots {\otimes }_{{\alpha }^{\prime }}{P}_l:={\left(P_1^{\circ }{\otimes }_{\alpha }\cdots {\otimes }_{\alpha }{P}_l^{\circ }\right)}^{\circ }.$$

$${\left(P_1^{\circ }{\otimes }_{ϵ}\cdots {\otimes }_{ϵ}{P}_l^{\circ }\right)}^{\circ }\subseteq {\left(P_1^{\circ }{\otimes }_{\alpha }\cdots {\otimes }_{\alpha }{P}_l^{\circ }\right)}^{\circ }\subseteq {\left(P_1^{\circ }{\otimes }_{\pi }\cdots {\otimes }_{\pi }{P}_l^{\circ }\right)}^{\circ }.$$

$$P_1{\otimes }_{\pi }\cdots {\otimes }_{\pi }{P}_l\subseteq P_1{\otimes }_{{\alpha }^{\prime }}\cdots {\otimes }_{{\alpha }^{\prime }}{P}_l\subseteq P_1{\otimes }_{ϵ}\cdots {\otimes }_{ϵ}{P}_l.$$

$$T_1^t\otimes \cdots \otimes T_l^t\left(Q_1^{\circ }{\otimes }_{\alpha }\cdots {\otimes }_{\alpha }{Q}_l^{\circ }\right)\subseteq P_1^{\circ }{\otimes }_{\alpha }\cdots {\otimes }_{\alpha }{P}_l^{\circ }.$$

$$T_1\otimes \cdots \otimes T_l\left(P_1{\otimes }_{{\alpha }^{\prime }}\cdots {\otimes }_{{\alpha }^{\prime }}{P}_l\right)\subseteq Q_1{\otimes }_{{\alpha }^{\prime }}\cdots {\otimes }_{{\alpha }^{\prime }}{Q}_l.$$

$$\left(P_1\cap M_1\right){\otimes }_{\alpha }\cdots {\otimes }_{\alpha }\left(P_l\cap M_l\right)=\left(P_1{\otimes }_{\alpha }\cdots {\otimes }_{\alpha }{P}_l\right)\cap {\otimes }_{i=1}^l{M}_i.$$

$$T_1\otimes \cdots \otimes T_l\left(P_1{\otimes }_{\alpha }\cdots {\otimes }_{\alpha }{P}_l\right)=T_1{P}_1{\otimes }_{\alpha }\cdots {\otimes }_{\alpha }{T}_l{P}_l.$$

$$\begin{gathered} \left(P_1\cap M_1\right){\otimes }_{{\alpha }^{\prime }}\cdots {\otimes }_{{\alpha }^{\prime }}\left(P_l\cap M_l\right)={\left({\left(P_1\cap M_1\right)}^{\diamond }{\otimes }_{\alpha }\cdots {\otimes }_{\alpha }{\left(P_l\cap M_l\right)}^{\diamond }\right)}^{\diamond }= \\ (L_1(P_1^{\circ }){\otimes }_{\alpha }\cdots {\otimes }_{\alpha }{L}_l(P_1^{\circ }){)}^{\diamond }=(L_1\otimes \cdots \otimes L_l\left(P_1^{\circ }{\otimes }_{\alpha }\cdots {\otimes }_{\alpha }{P}_l^{\circ }\right){)}^{\diamond }= \\ ({\left(P_1^{\circ }{\otimes }_{\alpha }\cdots {\otimes }_{\alpha }{P}_l^{\circ }\right)}^{\circ }\cap {\otimes }_{j=1}^l{M}_j{)}^{\diamond \diamond }=P_1{\otimes }_{{\alpha }^{\prime }}\cdots {\otimes }_{{\alpha }^{\prime }}{P}_l\cap {\otimes }_{j=1}^l{M}_j. \end{gathered}$$

Full text

*Keywords: Convex body, Tensor product of convex sets, Tensor product of Banach spaces, Hilbertian tensor norm, Ideals of linear operators, Grothendieck’s inequality.

2000 Mathematics Subject Classification: 46M05, 52A21, 47L20, 15A69.

1. Introduction

Given two compact convex sets $K_{1}$ and $K_{2}$, there are several different manners to construct a new compact convex set that can be considered as a tensorial product of them. Some of these notions were studied in relation with the so-called Choquet Theory, as done by Z. Semadeni in [26], I. Namioka and R.R. Phelps in [20] or E. Behrends and G. Wittstock in [4]. More recently, tensor products of convex sets have been studied in relation with the so called quantum information theory, as can be found in [2] by G. Aubrun and S. Szarek. Other classes of tensor products as well as some properties of those already mentioned can be found in [5, 12, 18, 28].

In a previous paper, we characterized when a centrally symmetric convex body $B$ in the Euclidean space $\mathbb{R}^{d}$ of dimension $d=d_{1}\cdots d_{l}$, is the unit ball of a Banach space whose norm is a reasonable crossnorm (see [10, Theorem 3.2]). That is, we showed how to determine in purely geometric terms, if $B$ is the unit ball of a reasonable crossnorm on a tensor product space $\otimes_{i=1}^{l}\left(\mathbb{R}^{d_{i}},\|\cdot\|_{i}\right),$ when the norms on each factor are not determined a priori. In this same line, we develop here a theory of tensor products of centrally symmetric convex bodies, which is consistent with the theory of tensor norms on finite dimensional Banach spaces. Our main result is that there is a bijection between both, tensor norms on finite dimensions and tensor products of [math]-symmetric convex bodies. Moreover, this bijection preserves duality, injectivity and projectivity (Theorem 3.8).

The theory of tensor norms was mainly developed by A. Grothendieck [11]. It is a fundamental tool in the modern study of Banach spaces. Among other things, it establishes the foundations of the theory of ideals of linear operators and the local theory of Banach spaces, as can bee seen in [6, 8, 7, 24, 27]. Its influence extends to a wide range of areas from Mathematical Analysis to Graph Theory or Computer Science [9, 17, 23]. When the spaces are finite dimensional, Theorem 3.8 translates this tool into the context of Convex Geometry. Indeed, a geometric formulation of Grothendieck‘s Theorem is provided in Section 4.

We now briefly expose the contents of the paper. We begin by developing a theory of tensor products of convex bodies with [math] in the interior. In this part (Section 2) we do not assume that the sets are centrally symmetric. In this study, two particular tensor products play a fundamental role, namely the projective and the injective tensor products $\otimes_{\pi},\,\otimes_{\epsilon}$. Their main properties, that will be used frequently afterwards, are stated.

If $P_{i}\subset\mathbb{E}_{i}$, $\dim\mathbb{E}_{i}={d_{i}},$ $i=1,...,l,$ are convex bodies with zero in the interior, we say that a compact convex set $Q\subset\mathbb{R}^{d_{1}\cdots d_{l}}$ is a tensorial convex body with respect to $P_{i}$, $i=1,\ldots,l,$ if $P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}\subseteq P\subseteq P_{l}\otimes_{\epsilon}\cdots\otimes_{\epsilon}P_{l}.$ A tensor product of convex bodies $\otimes_{\alpha}$ (Definition 2.6) assigns to each $l$-tuple as before, a tensorial convex body with respect to $P_{i}$, $P_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}P_{l}\subset\otimes_{i=1}^{l}\mathbb{\mathbb{E}}_{i}$ that satisfies the following uniform property: For every linear mapping $T_{i}:\mathbb{E}_{i}\rightarrow\mathbb{F}_{i},$ $i=1,..,l.$ If $P_{i}\subset\mathbb{E}_{i}$, $Q_{i}\subset\mathbb{F}_{i},$ $i=1,\ldots,l,$ are convex bodies with [math] in the interior and $T_{i}\left(P_{i}\right)\subseteq Q_{i},$ then

[TABLE]

An important fact derived from such uniform property is that tensor products of convex bodies are invariant under the tensor product of linear isomorphisms (see Proposition 2.7). This also implies that, in the case of [math]-symmetric convex bodies, these tensor products are continuous with respect to the Banach-Mazur distance (Proposition 3.5).

In Definiton 2.9, we introduce duality of tensor products by means of polarity. It holds that the dual $\otimes_{\alpha^{\prime}}$ of a tensor product of convex bodies $\otimes_{\alpha}$ is a tensor product of convex bodies, as well. It satisfies $\otimes_{\alpha^{\prime\prime}}=\otimes_{\alpha}.$

The injective and the projective properties of a tensor product of convex bodies are stated in terms of preservation under taking sections and quotients, respectively (see Definition 2.11). The fundamental relation between tensor products of convex bodies and their duals is exhibited in Theorem 2.12, where we prove that duals of projective tensor products of convex bodies are injective tensor products of convex bodies and vice versa.

In Section 3 is where we apply these results to develop a theory of tensor products of [math]-symmetric convex bodies. It is in this context where we state our main result, Theorem 3.8, which provides the existence of a bijection between tensor products of [math]-symmetric convex bodies and tensor norms on finite dimensional spaces. It widens the scope of the multilinear generalization of Minkowski‘s bijection between norms and centrally symmetric convex bodies, established in Theorem 3.2 and Corollary 3.4 of [10].

We study some important geometric aspects of these tensor products, as is the case of the Banach-Mazur distance and the injective and the projective properties. In Subsection 3.4 que provide explicit representations of the biggest injective and the smallest projective tensor products. In Subsection 3.5 we study the Löwner and John ellipsoids associated to a tensor product of [math]-symmetric convex bodies. We also prove that tensor products of convex bodies with enough symmetries have enough symmetries, too (Proposition 3.16).

The results presented so far are valid in the case of tensor products of any fixed number of spaces $l\in\mathbb{N}$. In the case of products of two factors, there is also a bijection with the ideals of linear operators preserving duality, injectivity and projectivity (see Corollary 4.1). This is studied in Section 4, where we develop in more detail the Hilbertian tensor product $\omega_{2}$, and give the geometric formulation of Grothendieck‘s Theorem, see (4.4).

1.1. Notation and preliminaires

Throughout this paper the letters $\mathbb{E},\mathbb{F}$ or $\mathbb{E}_{i},\mathbb{F}_{i},$ will denote Euclidean spaces over $\mathbb{R}$. The scalar product on $\mathbb{E}$ will be denoted by $\left\langle\cdot,\cdot\right\rangle_{\mathbb{E}}$ and the associated Euclidean ball by $B_{\mathbb{E}}.$ Given a linear map $T:\mathbb{E}\rightarrow\mathbb{F},$ its transpose $T^{t}$ is the linear map such that $\left\langle Tx,y\right\rangle_{\mathbb{F}}=\left\langle x,T^{t}y\right\rangle_{\mathbb{E}}.$

Given vector spaces $V_{1},\cdots,V_{l},W$ over the same field ($\mathbb{R}$ or $\mathbb{C}$) and a multilinear mapping $T:V_{1}\times\cdots\times V_{l}\rightarrow W$, we will denote $\hat{T}:\otimes_{i=1}^{l}V_{i}\rightarrow W$ the unique linear mapping such that $T=\hat{T}\circ\otimes$, where $\otimes$ denotes the canonical multilinear mapping:

[TABLE]

When the spaces $\mathbb{E}_{i}$, $i=1,\ldots,l$, are Euclidean spaces, $\otimes_{i=1}^{l}\mathbb{E}_{i}$ has a natural scalar product: it is defined in decomposable vectors as

[TABLE]

and extended to $\otimes_{i=1}^{l}\mathbb{E}_{i}$ by multilinearity. The space $\otimes_{H,i=1}^{l}\mathbb{E}_{i}:=\left(\otimes_{i=1}^{l}\mathbb{E}_{i},\left\langle\cdot,\cdot\right\rangle_{H}\right)$ is called the Hilbert tensor product of $\mathbb{E}_{1},\ldots,\mathbb{E}_{l}$ (the details can be seen in [15, Section 2.6]). Unless otherwise is stated, we will always assume that this is the euclidean structure defined in the tensor product of euclidean spaces.

Every compact convex set $P\subset\mathbb{E}$ with nonempty interior, $\text{int}(P)\neq\emptyset$, is called a convex body. In addition, if $P=-P$ then $P$ is called a [math]-symmetric convex body. We write $\mathcal{B}\left(\mathbb{E}\right)$ to denote the set of [math]-symmetric convex bodies contained in $\mathbb{E}.$

Given a nonempty set $C\subset\mathbb{E}$, its polar set is $C^{\circ}:=\left\{y\in\mathbb{E}:{\sup}_{x\in C}\left\langle x,y\right\rangle_{\mathbb{E}}\leq 1\right\}.$ The Minkowski functional (or gauge function) of $P\in\mathcal{B}\left(\mathbb{E}\right)$ is defined as

[TABLE]

A fundamental result concerning [math]-symmetric convex bodies is the bijection between norms defined on $\mathbb{E}$ and [math]-symmetric convex bodies, given by the Minkowski functional. This result, originally due to H. Minkowksi [19], establishes that the map

[TABLE]

is a bijection, the unit ball of the space $\left(\mathbb{E},g_{P}\right)$ is $P$ and $g_{P^{\circ}}\left(x\right)=\left\|\left\langle\cdot,x\right\rangle:\left(\mathbb{E},g_{p}\right)\rightarrow\mathbb{R}\right\|.$

The proof of the previous result as well as the fundamentals about convex bodies and Convex Geometry that will be used in this paper can be found in [25].

Recall that whenever $X_{1},\ldots,X_{l}$ are Banach spaces, the projective tensor norm $\pi$ and the injective tensor norm $\epsilon$ on $\otimes_{i=1}^{l}{X}_{i}$ are defined as:

[TABLE]

and

[TABLE]

for $u\in\otimes_{i=1}^{l}X_{i}.$

A norm $\alpha\left(\cdot\right)$ on the tensor product $\otimes_{i=1}^{l}X_{i}$ is a reasonable crossnorm if

[TABLE]

Both the projective and the injective tensor norms are reasonable crossnorms. Clearly, they are, respectively, the biggest and the smallest reasonable crossnorm. We suggest the monographs [6, 24] for a thorough treatment of tensor products of Banach spaces as well as for the fundamental properties of the projective and the injective tensor norms.

2. Tensorial convex bodies and tensor products of convex bodies

In the theory of tensor norms there is a significant distinction between the notions of reasonable crossnorm and of tensor norm. The first one refers to a norm on the tensor product of some fixed Banach spaces for which (1.1) holds. The second refers to a norm that, in addition, can be defined on the tensor product of any Banach spaces and such that it preserves the continuity of some mappings (the so called uniform property, see e.g. [7, pp.10]). This subtle distinction is also present when considering products of convex sets (compare Definitions 2.5 and 2.6 below).

In this section the convex sets will be fixed and they are not assumed to be [math]-symmetric.

2.1. The projective and the injective tensor product of convex bodies

Given $l$ convex bodies containing [math] in the interior, $P_{i}\subset\mathbb{E}_{i},$ $i=1,\ldots,l,$ their projective tensor product is defined as the following convex body in $\otimes_{i=1}^{l}\mathbb{E}_{i}$:

[TABLE]

In this case, [math] is also in the interior of $P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}.$ Their** injective tensor product** is the convex body in $\otimes_{i=1}^{l}\mathbb{E}_{i}$ defined as:

[TABLE]

Here, the polarity is the one determined by $\left\langle\cdot,\cdot\right\rangle_{H}.$ It holds that $0\in\text{int}(P_{1}\otimes_{\epsilon}\cdots\otimes_{\epsilon}P_{l})$, since such property is preserved by taking polars [25, Theorem 1.6.1.].

These products will play a fundamental role in what follows. They were introduced in [2] and [3, Section 4.1], respectively.

The projective tensor product satisfies a universal property, that we now describe. Let us consider $l$ convex bodies with [math] in the interior, $P_{i}\subset\mathbb{E}_{i}$ $i=1,\ldots,l.$ We say that the pair $(P,\varphi)$ of a convex body $P\subset\mathbb{E},$ $0\in\text{int}(P),$ and a multilinear mapping $\varphi:\mathbb{E}_{1}\times\cdots\times\mathbb{E}_{l}\rightarrow\mathbb{E}$ such that $\varphi\left(P_{1},...,P_{l}\right)\subseteq P$ has property ($\mathcal{UP}$) if:

For every convex body $Q\subset\mathbb{F}$ with $0\in\text{int}(Q)$ and every multilinear mapping $T:\mathbb{E}_{1}\times\cdots\times\mathbb{E}_{l}\rightarrow\mathbb{F},$ if $T\left(P_{1},...,P_{l}\right)\subseteq Q,$ there exists a unique linear mapping $T_{\varphi}:\mathbb{E}\rightarrow\mathbb{F}$ such that $T_{\varphi}\left(P\right)\subseteq Q$ and $T_{\varphi}\circ\varphi=T.$

Proposition 2.1**.**

(Universal property of $\otimes_{\pi}$) Let $P_{i}\subset\mathbb{E}_{i},$ $i=1,\ldots,l,$ be convex bodies containing [math] in the interior. Then the pair ($P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}$, $\otimes$) satisfies ($\mathcal{UP}$).

Proof.

By definition of $\otimes_{\pi},$ $\left\{x^{1}\otimes\cdots\otimes x^{l}\in\otimes_{i=1}^{l}\mathbb{E}_{i}:x^{i}\in P_{i}\right\}\subseteq P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}.$ Now, let $T:\mathbb{E}_{1}\times\cdots\times\mathbb{E}_{l}\rightarrow\mathbb{F}$ be a multilinear mapping such that $T\left(P_{1},...,P_{l}\right)\subseteq Q$ and let $\hat{T}$ be its associated linear mapping. The relation $\hat{T}\circ\otimes=T$ implies that $\hat{T}\left(P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}\right)\subseteq Q$. ∎

Proposition 2.2**.**

*(Uniqueness for the universal property)

Let $P_{i}\subset\mathbb{E}_{i},$ $i=1,...,l$ be convex bodies containing [math] in the interior. If $P\subset\mathbb{E}$ is a convex body with [math] in its interior, $\varphi:\mathbb{E}_{1}\times\cdots\times\mathbb{E}_{l}\rightarrow\mathbb{E}$ is a multilinear map such that $\varphi\left(P_{1},...,P_{l}\right)\subseteq P$ and the pair $\left(P,\varphi\right)$ satisfies ($\mathcal{UP}$), then there exists a linear isomorphism $\Psi:\mathbb{E}\rightarrow\otimes_{i=1}^{l}\mathbb{E}_{i}$ such that $\Psi\left(P\right)=P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}$ and $\Psi\varphi=\otimes.$*

Proof.

Since $\left(P,\varphi\right)$ and $(P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l},\otimes)$ have the property ($\mathcal{UP}$), there exist linear mappings $S:\mathbb{E}\rightarrow\otimes_{i=1}^{l}\mathbb{E}_{i},$ and $T:\otimes_{i=1}^{l}\mathbb{E}_{i}\rightarrow\mathbb{E}$ such that $S\circ\varphi=\otimes,$ $T\circ\otimes=\varphi,$ $S\left(P\right)\subseteq P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}$ and $T\left(P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}\right)\subseteq P.$ From this, the linear map $TS:\mathbb{E}\rightarrow\mathbb{E}$ verifies: $TS\varphi\left(x^{1},...,x^{l}\right)=\varphi\left(x^{1},...,x^{l}\right),$ for every $x^{i}\in\mathbb{E}_{i}.$ This equality, together with the uniqueness of the linear extension, implies that $TS=I_{\mathbb{E}}$. Then, $P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}=ST(P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l})\subseteq S\left(P\right)\subseteq P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}.$ Thus, $S\left(P\right)=P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}$ and $S\circ\varphi=\otimes.$ This completes the proof. ∎

The next property of $\otimes_{\pi}$ and $\otimes_{\epsilon}$ will be fundamental to what follows:

Proposition 2.3**.**

(Uniform property of $\otimes_{\pi},\otimes_{\epsilon}$) Let $T_{i}:\mathbb{E}_{i}\rightarrow\mathbb{F}_{i},$ $i=1,\ldots,l,$ be linear maps. If $P_{i}\subset\mathbb{E}_{i}$, $Q_{i}\subset\mathbb{F}_{i}$ are convex bodies containing [math] in the interior and $T_{i}\left(P_{i}\right)\subseteq Q_{i},$ $i=1,\ldots,l,$ then

[TABLE]

and

[TABLE]

Proof.

To prove the first inclusion, observe that the linearity of $T_{1}\otimes\cdots\otimes T_{l}$ and the definition of $\otimes_{\pi}$ imply:

[TABLE]

This equality, together with $T_{i}\left(P_{i}\right)\subseteq Q_{i}$, yield to the first inclusion. To prove the second, we use that $T_{i}\left(P_{i}\right)\subseteq Q_{i}$ implies $T_{i}^{t}\left(Q_{i}^{\circ}\right)\subseteq P_{i}^{\circ}.$ Hence, by the definition of $\otimes_{\epsilon},$ for every $z\in P_{1}\otimes_{\epsilon}\cdots\otimes_{\epsilon}P_{l}$ and $y^{i}\in Q_{i}^{\circ}$ we have that $\left|\left\langle z,T_{1}^{t}y^{1}\otimes\cdots\otimes T_{l}^{t}y^{l}\right\rangle_{H}\right|\leq 1$ or, equivalently, $\left|\left\langle\left(T_{1}\otimes\cdots\otimes T_{l}\right)z,y^{1}\otimes\cdots\otimes y^{l}\right\rangle_{H}\right|\leq 1.$ Thus, $\left(T_{1}\otimes\cdots\otimes T_{l}\right)z\in Q_{1}\otimes_{\epsilon}\cdots\otimes_{\epsilon}Q_{l}$ and the proof is completed. ∎

The following result sets the injectivity of $\otimes_{\epsilon}$ and the projectivity of $\otimes_{\pi}$ as defined below, in Definition 2.11.

Proposition 2.4**.**

Let $P_{i}\subset\mathbb{E}_{i},$ $i=1,\ldots,l,$ be convex bodies with $0\in int(P_{i})$.

1. (1)

For each subspaces $M_{i}\subset\mathbb{E}_{i}$,

[TABLE] 2. (2)

For any $l$-tuple of surjective linear maps $T_{i}:\mathbb{E}_{i}\rightarrow\mathbb{F}_{i},$

[TABLE]

Proof.

That $\otimes_{\epsilon}$ respects sections follows by applying the Hanh-Banach Theorem. By duality, the projective tensor product of convex bodies $\otimes_{\pi}$ preserves quotients. ∎

In the case of [math]-symmetric convex bodies $P_{i}\subset\mathbb{E}_{i},$ $i=1,\ldots,l,$ the products $\otimes_{\pi},\otimes_{\epsilon}$ and the norms $\pi(\cdot),\epsilon(\cdot)$ are related by the following equalities (see [10, Section 2]):

[TABLE]

The use of the terminology “projective tensor product” and “injective tensor product” for the class of convex bodies with [math] in the interior is, therefore, consistent with their use for tensor norms.

2.2. Tensorial convex bodies: definition and examples.

Definition 2.5**.**

A compact convex set $Q\subset\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}$ is called a tensorial convex body in $\mathbf{\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}}$ if there exist convex bodies $Q_{i}\subset\mathbb{R}^{d_{i}},$ $i=1,...,l,$ containing zero in the interior such that

[TABLE]

In this case, we say that $Q$ is a tensorial convex body with respect to $\mathbf{Q_{1},\ldots,Q_{l}}.$

Notice that since $0\in\text{int}(P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}),$ $P$ is also a convex body containing zero in the interior. Furthermore, as a consequence of Proposition 2.8 its polar set is also a tensorial convex body. The following are examples of tensorial convex bodies. Details can be found in [10].

1. (1)

The injective and the projective tensor product. Let $P_{i}\subset\mathbb{E}_{i},$ $i=1,\ldots,l,$ be convex bodies containing [math] in the interior. Then, trivially, $P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}$ and $P_{1}\otimes_{\epsilon}\cdots\otimes_{\epsilon}P_{l}$ satisfy (2.5) with respect to $P_{1},\ldots P_{l}$. Consequently, they are tensorial convex bodies. 2. (2)

Hilbertian tensor product of ellipsoids $\otimes_{2}$. If $\mathcal{E}_{i}=T_{i}\left(B_{2}^{d_{i}}\right),$ $i=1,\ldots,l$ are ellipsoids in $\mathbb{R}^{d_{i}},$ $i=1,...,l$ respectively, then the Hilbertian tensor product of $\mathcal{E}_{1},\ldots,\mathcal{E}_{l}$, introduced in [2], is defined as

[TABLE]

It can be directly proved that $\mathcal{E}_{1}\otimes_{2}\cdots\otimes_{2}\mathcal{E}_{l}$ is the closed unit ball of the Hilbert tensor product $\otimes_{H,i=1}^{l}\left(\mathbb{R}^{d_{i}},g_{\mathcal{E}_{i}}\right)$. 3. (3)

Unit ball of $\ell_{p}^{d}$. Let $d=d_{1}\cdots d_{l}$ and $1\leq p\leq\infty$. Then, $B_{p}^{d}=B_{p}^{d_{1},...,d_{l}}$ is a tensorial convex body in $\otimes_{i=1}^{l}\mathbb{R}^{d_{i}}$.

2.3. Tensor product of convex bodies with zero in the interior

In what follows we will fix the scalar product on $\otimes_{i=1}^{l}\mathbb{E}_{i}$ as the one associated to the Hilbert tensor product $\otimes_{H,i=1}^{l}\mathbb{E}_{i}.$ In this way, if $Q\subset\otimes_{i=1}^{l}\mathbb{E}_{i}$ is a nonempty subset, its polar set is

[TABLE]

Definition and basic properties of tensor products

Definition 2.6**.**

A tensor product $\otimes_{\alpha}$ of order $l$ of convex bodies with zero in the interior is an assignment of a convex body $P_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}P_{l}\subset\otimes_{i=1}^{l}\mathbb{\mathbb{E}}_{i}$, to each $l$-tuple $P_{i}\subset\mathbb{E}_{i},$ $i=1,...,l,$ of convex bodies containing [math] in the interior, such that the following conditions are satisfied:

1. (1)

$P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}\subseteq P_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}P_{l}\subseteq P_{1}\otimes_{\epsilon}\cdots\otimes_{\epsilon}P_{l},$ that is, it is a tensorial convex body with respect to $P_{1},\ldots,P_{l}$. 2. (2)

(Uniform property) For every linear mapping $T_{i}:\mathbb{E}_{i}\rightarrow\mathbb{F}_{i},$ $i=1,..,l$, if $Q_{i}\subset\mathbb{F}_{i},$ are convex bodies with [math] in the interior and $T_{i}\left(P_{i}\right)\subseteq Q_{i},$ then

[TABLE]

When no confusion can arise, we will refer to them simply as tensor products of convex bodies. Note that $0\in\text{int}(P_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}P_{l})$. Since condition (1) holds for $\otimes_{\pi},\otimes_{\epsilon},$ (see Subsection 2.1), Proposition 2.3 implies that they are tensor products of convex bodies.

Proposition 2.7**.**

Let $\otimes_{\alpha}$ be a tensor product of convex bodies with zero in the interior and let $P_{i}\subset\mathbb{E}_{i},$ $i=1,\ldots,l,$ be convex bodies with $0\in int(P_{i})$. Then, for every bijective linear mapping $T_{i}:\mathbb{E}_{i}\rightarrow\mathbb{F}_{i}$ we have:

[TABLE]

Proof.

Observe that being $T_{i}:\mathbb{E}_{i}\rightarrow\mathbb{F}_{i}$ bijective, we know that $Q_{i}=T_{i}\left(P_{i}\right)$ is a convex body containing [math] in the interior. By the uniform property of $\otimes_{\alpha},$ we have $T_{1}\otimes\cdots\otimes T_{l}\left(P_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}P_{l}\right)\subseteq T_{1}P_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}T_{l}P_{l}$ and $T_{1}^{-1}\otimes\cdots\otimes T_{l}^{-1}\left(Q_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}Q_{l}\right)\subseteq T_{1}^{-1}Q_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}T_{l}^{-1}Q_{l}.$ Since $T_{i}^{-1}(Q_{i})=P_{i}$, we have that $T_{1}\otimes\cdots\otimes T_{l}\left(P_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}P_{l}\right)=T_{1}P_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}T_{l}P_{l}.$ ∎

Polarity on tensor products of convex bodies

The relation of polarity between $\otimes_{\epsilon}$ and $\otimes_{\pi}$ determined by the definition of $\otimes_{\epsilon}$ can be formulated in the following equivalent ways:

Proposition 2.8**.**

Let $P_{i}\subset\mathbb{E}_{i},$ $i=1,\ldots,l$ be convex bodies with $0\in\text{int}(P_{i})$. Then,

1. (1)

$(P_{1}\otimes_{\epsilon}\cdots\otimes_{\epsilon}P_{l})^{\circ}=P_{1}^{\circ}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}^{\circ}.$** 2. (2)

$(P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l})^{\circ}=P_{1}^{\circ}\otimes_{\epsilon}\cdots\otimes_{\epsilon}P_{l}^{\circ}.$**

Thus, according to the following definition, $\otimes_{\epsilon}$ and $\otimes_{\pi}$ are dual tensor products:

Definition 2.9**.**

Let $\otimes_{\alpha}$ be a tensor product of convex bodies with zero in the interior. Its ** dual tensor product** $\boldsymbol{\otimes_{\alpha^{\prime}}}$ is the tensor product that assigns, to each $l$-tuple $P_{i}\subset\mathbb{E}_{i},$ $i=1,...,l,$ of convex bodies with $0\in int(P_{i})$, the convex body

[TABLE]

Now we check that $\boldsymbol{\otimes_{\alpha^{\prime}}}$ is well defined. Since $0\in P_{1}^{\circ}\otimes_{\alpha}\cdots\otimes_{\alpha}P_{l}^{\circ}$, then $0\in(P_{1}^{\circ}\otimes_{\alpha}\cdots\otimes_{\alpha}P_{l}^{\circ})^{\circ}$ (see [25, Theorem 1.6.1.]). Condition (1) in Definition 2.6 follows using such condition for $P_{1}^{\circ}\otimes_{\alpha}\cdots\otimes_{\alpha}P_{l}^{\circ}$. Then, by taking polars the following inclusions are obtained:

[TABLE]

By Proposition 2.8, this is the same as:

[TABLE]

To prove condition (2), consider $T_{i}:\mathbb{E}_{i}\rightarrow\mathbb{F}_{i}$ such that $T_{i}\left(P_{i}\right)\subseteq Q_{i}$, $i=1,...,l$. Using the uniform property of $\otimes_{\alpha}$, we get:

[TABLE]

Thus, $\left(T_{1}^{t}\otimes\cdots\otimes T_{l}^{t}\right)^{t}\left(P_{1}^{\circ}\otimes_{\alpha}\cdots\otimes_{\alpha}P_{l}^{\circ}\right)^{\circ}\subset\left(Q_{1}^{\circ}\otimes_{\alpha}\cdots\otimes_{\alpha}Q_{l}^{\circ}\right)^{\circ}.$ That is,

[TABLE]

∎

The following proposition collects the basic properties of $\otimes_{\alpha^{\prime}}.$ We do not include its proof, since it is a direct consequence of the definition of $\otimes_{\alpha^{\prime}}$ and Proposition 2.8.

Proposition 2.10**.**

The following hold:

1. (1)

The dual of $\otimes_{\pi}$ (resp. $\otimes_{\epsilon}$) is $\otimes_{\epsilon}$ (resp. $\otimes_{\pi}$). 2. (2)

If $\otimes_{\alpha}$ is a tensor product of convex bodies, then $\otimes_{\alpha^{\prime\prime}}=\otimes_{\alpha}.$ 3. (3)

If $\otimes_{\alpha},\otimes_{\beta}$ are tensor products of convex bodies such that $P_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}P_{l}\subseteq P_{1}\otimes_{\beta}\cdots\otimes_{\beta}P_{l}$ for all convex bodies $P_{i}\subset\mathbb{E}_{i},$ $0\in\text{int}(P_{i}),$ then $P_{1}\otimes_{\beta^{\prime}}\cdots\otimes_{\beta^{\prime}}P_{l}\subseteq P_{1}\otimes_{\alpha^{\prime}}\cdots\otimes_{\alpha^{\prime}}P_{l}.$

Sections and quotients: injective and projective properties

Definition 2.11**.**

We say that a tensor product of convex bodies with zero in the interior $\otimes_{\alpha}$ is injective if for each $P_{i}\subset\mathbb{E}_{i}$ with $0\in\text{int}(P_{i})$ and each subspace $M_{i}\subseteq\mathbb{E}_{i},$ $i=1,...,l,$ it holds that

[TABLE]

Here, the scalar product on the space $M_{i}$ is the one induced by $\mathbb{E}_{i}.$ In this way each $P_{i}\cap M_{i}$ is a convex body with [math] in the interior.

We say that $\otimes_{\alpha}$ is projective if for each $P_{i}\subset\mathbb{E}_{i},$ $0\in\text{int}(P_{i}),$ and every surjective linear mapping $T_{i}:\mathbb{E}_{i}\rightarrow\mathbb{F}_{i},$ $i=1,...,l,$ it holds that

[TABLE]

Injectivity and projectivity are dual properties, in the following sense:

Theorem 2.12**.**

Let $\otimes_{\alpha}$ be a tensor product of convex bodies, then $\otimes_{\alpha}$ is projective if and only if $\otimes_{\alpha^{\prime}}$ is injective.

Proof.

Suppose that $\otimes_{\alpha}$ is projective. For each $j$, let $M_{j}\subset\mathbb{E}_{j}$ be a subspace and let $L_{j}$ be the orthogonal projection onto $M_{j}.$ For each $P_{j}\subset\mathbb{E}_{j},$ $0\in\text{int}(P_{j}),$ $\left(P_{j}\cap M_{j}\right)^{\diamond}$ denotes the polar set of $P_{j}\cap M_{j}\subseteq M_{j}$ determined by the scalar product on $M_{j}$ induced by $\mathbb{E}_{j}.$

Using the relation [3, (1.13)] several times and the projectivity of $\otimes_{\alpha}$, we have

[TABLE]

Consequently, $\otimes_{\alpha^{\prime}}$ is injective.

To prove the reciprocal, we will use the following relation: if $T:\mathbb{E}\rightarrow\mathbb{F}$ is a linear map and $P\subset\mathbb{E}$ is a convex set then,

[TABLE]

Let $T_{j}:\mathbb{E}_{j}\rightarrow\mathbb{F}_{j}$, $j=1,...,l,$ be surjective mappings. In such case, $T_{1}\otimes\cdots\otimes T_{l}$ is also surjective. The mappings $T_{j}^{t}$ and $(T_{1}\otimes\cdots\otimes T_{l})^{t}=T_{1}^{t}\otimes\cdots\otimes T_{l}^{t}$ are injective.

Let us assume that $\otimes_{\alpha^{\prime}}$ is injective. Using the injectivity of $\otimes_{\alpha^{\prime}}$ and applying (2.9), we have:

[TABLE]

(*) is due to the injectivity of $T_{1}^{t}\otimes\cdots\otimes T_{l}^{t}$ and Proposition 2.7. Indeed, we must have:

[TABLE]

Finally, the result follows directly by taking polars in the previous equality and using (2.6). ∎

Remark 2.13*.*

The projective tensor product of convex bodies $\otimes_{\pi}$, which is projective (see Proposition 2.4) is not injective. To see this, recall that there exist finite dimensional Banach spaces $M\subset E$, $N$ and an element in $\mathcal{L}(M;N^{*})$ such that each of its extensions in $\mathcal{L}(E;N^{*})$ has norm strictly greater [16, Theorem 4]. This means, by the Hahn-Banach theorem and by the fundamental relation $\mathcal{L}(E;N^{*})=(E\otimes_{\pi}N)^{*}$ [6, pp. 27], that $(B_{E}\cap M)\otimes_{\pi}B_{N}\neq(B_{E}\otimes_{\pi}B_{N})\cap M\otimes N,$ where $B_{E},$ $B_{M}$ are the closed unit balls of $E,N.$

By duality, we have that the injective tensor product of convex bodies is not projective.

The closest injective and projective tensor products

The following relation on tensor products defines an order: let $\otimes_{\alpha}$ and $\otimes_{\beta}$ be tensor products of convex bodies. We say that $\otimes_{\alpha}\subseteq\otimes_{\beta}$ if and only if for every tuple of convex bodies $P_{i}\subset\mathbb{E}_{i},$ $i=1,\ldots,l,$ containing [math] in their interior, $P_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}P_{l}\subseteq P_{1}\otimes_{\beta}\cdots\otimes_{\beta}P_{l}.$ We say that $\otimes_{\alpha}=\otimes_{\beta}$ if and only if $\otimes_{\alpha}\subseteq\otimes_{\beta}$ and $\otimes_{\beta}\subseteq\otimes_{\alpha}$ hold. In this order, given any tensor product of convex bodies $\otimes_{\alpha}$, it holds that $\otimes_{\pi}\subseteq\otimes_{\alpha}\subseteq\otimes_{\epsilon}$.

Given a tensor product of convex bodies $\otimes_{\alpha}$, let $\otimes_{\alpha^{inj}}$ be defined on each tuple of convex bodies $P_{i}\subset\mathbb{E}_{i},$ $0\in\text{int}(P_{i}),$ $i=1,\ldots,l,$ as:

[TABLE]

Proposition 2.14**.**

Let $\otimes_{\alpha}$ be a tensor product of convex bodies, then $\otimes_{\alpha^{inj}}$ is the smallest injective tensor product of convex bodies such that $\otimes_{\alpha}\subseteq\otimes_{\alpha^{inj}}$.

Proof.

Observe first that $\otimes_{\alpha^{inj}}$ is well defined, since $\otimes_{\alpha}\subseteq\otimes_{\epsilon}$ and $\otimes_{\epsilon}$ is injective. Let $P_{i}\subset\mathbb{E}_{i},$ $0\in\text{int}(P_{i}),$ $i=1,\ldots,l,$ be any $l$-tuple of convex bodies. Then, $P_{1}\otimes_{\alpha^{inj}}\cdots\otimes_{\alpha^{inj}}P_{l}$ is a compact convex set such that

[TABLE]

Since $0\in int(P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l})$, the same holds for $P_{1}\otimes_{\alpha^{inj}}\cdots\otimes_{\alpha^{inj}}P_{l}.$ The uniform property (condition $(2)$ in Definition 2.6) is satisfied because each $\otimes_{\beta}$ satisfies it.

To see that $\otimes_{\alpha^{inj}}$ is injective, consider any subspaces $M_{i}\subseteq\mathbb{E}_{i},$ $i=1,...,l,$. Then, using that each $\otimes_{\beta}$ is injective, we have:

[TABLE]

Finally, being $\otimes_{\alpha^{inj}}$ an injective tensor product, it has to be the smallest one bigger than $\otimes_{\alpha}.$ ∎

Given a tensor product of convex bodies $\otimes_{\alpha},$ let $\otimes_{\alpha^{proj}}$ be defined on each tuple of convex bodies $P_{i}\subset\mathbb{E}_{i},$ $0\in\text{int}(P_{i}),$ $i=1,\ldots,l,$ $\otimes_{\alpha^{proj}}$ as:

[TABLE]

Proposition 2.15**.**

For a given $\otimes_{\alpha}$, $\otimes_{\alpha^{proj}}$ is the biggest projective tensor product of convex bodies such that $\otimes_{\alpha^{proj}}\subseteq\otimes_{\alpha}$. Furthermore,

[TABLE]

Proof.

By Theorem 2.12 and Proposition 2.14, $\otimes_{\alpha^{proj}}$ is a projective tensor product of convex bodies. Then, from (2) and (3) in Proposition 2.10, it follows easily that $\otimes_{\alpha^{proj}}$ is the biggest projective tensor product of convex bodies below $\otimes_{\alpha}.$

To prove the second part, let $P_{i}\subset\mathbb{E}_{i},$ $i=1,\ldots,l$ be convex bodies containing [math] in the interior. Then, by Theorem 2.12, we have:

[TABLE]

*It is used that $(\cap_{i\in I}K_{i})^{\circ}=\overline{\text{conv}}\cup_{i\in I}K_{i}^{\circ}$ (see [25, Theorem 1.6.2]).

**It is used (2) in Proposition 2.10. ∎

3. Tensor products on finite dimensional Banach spaces and tensor products of [math]-symmetric convex bodies: a bijective correspondence.

We apply the results in the previous section to the case of [math]-symmetric convex bodies. The theory of tensor products thus derived is consistent with the theory of tensor products of finite dimensional Banach spaces. The precise statement is written in Theorem 3.8.

Definition 3.1**.**

A tensor product $\otimes_{\alpha}$ of order $l$ of [math]-symmetric convex bodies is an assignment of a [math]-symmetric convex body $P_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}P_{l}\subset\otimes_{i=1}^{l}\mathbb{\mathbb{E}}_{i}$, to each $l$-tuple $P_{i}\subset\mathbb{E}_{i},$ $i=1,...,l,$ of [math]-symmetric convex bodies, such that the following conditions are satisfied:

1. (1)

$P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}\subseteq P_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}P_{l}\subseteq P_{1}\otimes_{\epsilon}\cdots\otimes_{\epsilon}P_{l}.$ 2. (2)

(Uniform property) For every linear mapping $T_{i}:\mathbb{E}_{i}\rightarrow\mathbb{F}_{i},$ $i=1,..,l$, if $Q_{i}\subset\mathbb{F}_{i},$ are [math]-symmetric convex bodies and $T_{i}\left(P_{i}\right)\subseteq Q_{i},$ then

[TABLE]

where $\otimes_{\pi}$ and $\otimes_{\epsilon}$ are as in Section 2.1.

It can be directly checked that $\otimes_{\pi}$ and $\otimes_{\epsilon}$ satisfy Definition 3.1. It also follows from the following more general result:

Proposition 3.2**.**

Let $\otimes_{\alpha}$ be a tensor product of convex bodies. If at least one of the convex bodies with [math] in its interior $P_{i}\subset\mathbb{E}_{i}$, $i=1,\ldots,l$ is [math]-symmetric, then $P_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}P_{l}$ is a [math]-symmetric convex body.

Proof.

Assume, w.l.o.g. that $P_{1}\subset\mathbb{E}_{1}$, is [math]-symmetric. Consider the identity map on $\mathbb{E}_{i}$, $I_{i}=I_{\mathbb{E}_{i}}$ for $i=1,\ldots,l.$ By Proposition 2.7 $(-I_{1})\otimes\cdots\otimes I_{l}(P_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}P_{l})=(-P_{1})\otimes_{\alpha}\cdots\otimes_{\alpha}P_{l}.$ Since $P_{1}=-P_{1}$, we have $-(P_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}P_{l})=P_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}P_{l}$, as required. ∎

Corollary 3.3**.**

Every tensor product of convex bodies induces a tensor product of [math]-symmetric convex bodies.

3.1. Dual tensor product, injectivity and projectivity

Using Corollary 3.3, it follows that the notions and results of the previous section have analogues for [math]-symmetric convex bodies. We will use them, but we omit the proofs. Concretely, we say that $\otimes_{\alpha}$ is injective if it satisfies (2.7) and projective if it satisfies (2.8) for [math]-symmetric convex bodies.

Proposition 3.4**.**

Let $\otimes_{\alpha}$ be a tensor product of [math]-symmetric convex bodies. Then,

1. (1)

Relation (2.6) defines a tensor product of [math]-symmetric convex bodies $\boldsymbol{\otimes_{\alpha^{\prime}}}$. It will be called the dual tensor product of $\otimes_{\alpha}$. 2. (2)

$\otimes_{\alpha}$* is injective if and only if $\otimes_{\alpha^{\prime}}$ is projective.*

Proposition 2.10, as well as the definitions and results in Subsection 2.3 concerning injective $\otimes_{\alpha^{inj}}$ and projective $\otimes_{\alpha^{proj}}$ hulls, also have analogues.

3.2. Relation with the Banach-Mazur distance

We will see now that a tensor product of [math]-symmetric convex bodies is uniformly continuous with respect to the Banach-Mazur distance. Recall that the Banach-Mazur distance between [math]-symmetric convex bodies in a Euclidean space $\mathbb{E}$ is defined as:

[TABLE]

A complete exposition of the Banah-Mazur distance can be found in [27].

Proposition 3.5**.**

Let $P_{i},Q_{i}\subset\mathbb{E}_{i}$, $i=1,...,l$, be [math]-symmetric convex bodies. If $\otimes_{\alpha}$ is a tensor product convex bodies, then

[TABLE]

Proof.

Let us fix $i\in\left\{1,...,l\right\}$ and let $\lambda\geq\delta^{BM}\left(P_{i},Q_{i}\right)$. Then, there exists a linear isomorphism $T_{i}:\mathbb{E}_{i}\rightarrow\mathbb{E}_{i}$ such that $Q_{i}\subseteq T_{i}\left(P_{i}\right)\subseteq\lambda Q_{i}$. By Proposition 2.7, we have $P_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}T_{i}P_{i}\otimes_{\alpha}\cdots\otimes_{\alpha}P_{l}=I_{\mathbb{E}_{1}}\otimes\cdots\otimes T_{i}\otimes\cdots\otimes I_{\mathbb{E}_{l}}\left(P_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}P_{i}\otimes_{\alpha}\cdots\otimes_{\alpha}P_{l}\right).$ Therefore, if $S=I_{\mathbb{E}_{1}}\otimes\cdots\otimes T_{i}\otimes\cdots\otimes I_{\mathbb{E}_{l}}$ then

[TABLE]

Consequently,

[TABLE]

The result follows by iterating this relation along with the multiplicative triangle inequality of $\delta^{BM}$. ∎

3.3. Bijection with tensor norms

Recall that a tensor norm $\alpha$ of order** $l$** on the class of finite dimensional (abbreviated f.d.) Banach spaces assigns to each $l$-tuple $M_{1},\ldots,M_{l}$ of f.d. Banach spaces a norm $\alpha_{M_{1},\ldots,M_{l}}$ on the tensor product $\otimes_{i=1}^{l}M_{i}$ (simply denoted $\alpha$, and called a tensor norm) such that the two following conditions are satisfied:

1. (1)

$\alpha$ is a reasonable crossnorm. 2. (2)

$\alpha$ satisfies the uniform property i.e. for each $T_{i}\in\mathcal{L}\left(M_{i};N_{i}\right)$ $i=1,\ldots,l,$

[TABLE]

In relation with the so called Minkowski functional, it was seen in [10, Corollary 3.4] that a reasonable crossnorm corresponds to a tensorial body (see [10, Definition 3.3]) and vice versa.

Given a tensor norm $\alpha\left(\cdot\right)$ on $\otimes_{i=1}^{l}M_{i}$, $\otimes_{\alpha,i=1}^{l}M_{i}$ will denote the normed space $\left(\otimes_{i=1}^{l}M_{i},\alpha\right)$.

The dual tensor norm of $\alpha$, $\alpha^{\prime}$, is defined as follows: given any f.d. normed spaces $M_{1},\ldots,M_{l}$, for every $u\in\otimes_{i=1}^{l}M_{i},$

[TABLE]

A tensor norm $\alpha$ is called injective if for each subspace $E_{i}$ of a Banach space $X_{i},$ $\otimes_{\alpha,i=1}^{l}E_{i}$ is a subspace of $\otimes_{\alpha,i=1}^{l}X_{i}.$ A tensor norm $\alpha$ is called projective is for every $l$-tuple of quotient operators $S_{i}:X_{i}\rightarrow Y_{i}$, $S_{1}\otimes\cdots\otimes S_{l}:\otimes_{\alpha,i=1}^{l}X_{i}\rightarrow\otimes_{\alpha,i=1}^{l}Y_{i}$ is a quotient operator.

For a detailed exposition about tensor norms on Banach spaces, we suggest the reader the monographs [6, 7, 24].

Proposition 3.6**.**

Let $\otimes_{\alpha}$ be a tensor product of [math]-symmetric convex bodies. For every $l$-tuple $M_{i},$ $i=1,...,l,$ of f.d. Banach spaces, consider

[TABLE]

where $B_{i}$ is the closed unit ball of $M_{i}$. Then, $\left\|\cdot\right\|_{\otimes_{\alpha}}$ defines a tensor norm on the class of finite dimensional Banach spaces.

Proof.

Let $M_{i},$ $i=1,\ldots,l,$ be f.d. normed spaces and let fix a scalar product on each of them. By the discussion in Subsection 3.1 and (2.4), we have that (3.2) satisfies condition (1) above. That is, it is a reasonable crossnorm on $\otimes_{i=1}^{l}M_{i}$.

For $\left\|\cdot\right\|_{\otimes_{\alpha}}$ to be well defined on $\otimes_{i=1}^{l}M_{i}$, we must check that it does not depend on the scalar product we have considered on each $M_{i}$. To that end, let $\left[\cdot,\cdot\right]_{i},\left\langle\cdot,\cdot\right\rangle_{i}$ be scalar products on $M_{i}$, let $T_{i}:\left(M_{i},\left[\cdot,\cdot\right]_{i}\right)\rightarrow\left(M_{i},\left\langle\cdot,\cdot\right\rangle_{i}\right)$ denote the identity map and let $Q_{i}:=T_{i}B_{M_{i}}.$ From Proposition 2.7,

[TABLE]

Therefore, $g_{B_{M_{1}}\otimes_{\alpha}\cdots\otimes_{\alpha}B_{M_{l}}}=g_{Q_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}Q{}_{l}}.$ Consequently, $\left\|\cdot\right\|_{\otimes_{\alpha}}$ is a well defined reasonable crossnorm on the class of finite dimensional normed spaces.

To finish, we have to prove that this norm satisfies the uniform property for tensor norms. To see this, take $T_{i}\in\mathcal{L}\left(M_{i},N_{i}\right)$ such that $\left\|T_{i}\right\|\leq 1.$ Then, $T_{i}\left(B_{i}\right)\subseteq B_{N_{i}}.$ By the uniform property of $\otimes_{\alpha},$ we have

[TABLE]

This implies that $T_{1}\otimes\cdots\otimes T_{l}:\left(\otimes_{i=1}^{l}M_{i},\left\|\cdot\right\|_{\otimes_{\alpha}}\right)\rightarrow\left(\otimes_{i=1}^{l}N_{i},\left\|\cdot\right\|_{\otimes_{\alpha}}\right)$ has norm $\leq 1$. So, $\left\|\cdot\right\|_{\otimes_{\alpha}}$ is uniform, as required. ∎

Proposition 3.7**.**

Let $\|\cdot\|_{\alpha}$ be a tensor norm on f.d. spaces. For every $l$-tuple $P_{i}\subset\mathbb{E}_{i}$ $i=1,...,l,$ of [math]-symmetric convex bodies consider

[TABLE]

Then, $\otimes_{{\alpha}}$ defines a tensor product of [math]-symmetric convex bodies.

Proof.

Let $P_{i}\subset\mathbb{E}_{i}$ be [math]-symmetric convex bodies. Denote by $E_{i}=\left(\mathbb{\mathbb{E}}_{i},g_{P_{i}}\right)$ the normed space whose closed unit ball is $P_{i}$. Since $\|\cdot\|_{\alpha}$ is a tensor norm, then $B_{\otimes^{l}_{\|\cdot\|_{\alpha},i=1}\left(\mathbb{\mathbb{E}}_{i},g_{P_{i}}\right)}$ is a [math]-symmetric convex body for which $\epsilon\left(z\right)\leq\alpha\left(z\right)\leq\pi\left(z\right),\text{ for }z\in\otimes_{i=1}^{l}E_{i}.$

Thus, from (2.4) and (3.3), we have that $P_{1}\otimes_{\pi}\cdots\otimes_{\pi}P_{l}\subseteq P_{1}\otimes_{\|\cdot\|_{\alpha}}\cdots\otimes_{\|\cdot\|_{\alpha}}P_{l}\subseteq P_{1}\otimes_{\epsilon}\cdots\otimes_{\epsilon}P_{l}.$ The uniform property of $\otimes_{{\alpha}}$ follows directly from the uniform property of the norm $\alpha.$ ∎

Theorem 3.8**.**

Let $\otimes_{\alpha}$ and $\|\cdot\|_{\otimes_{\alpha}}$ be as in Propositions 3.6 and 3.7. Then, the mapping

[TABLE]

is a bijection. It holds that

1. i)

$\otimes_{\alpha^{\prime}}$* is dual to $\otimes_{\alpha}$ if and only if $\|\cdot\|_{\otimes_{\alpha^{\prime}}}$ is dual to $\|\cdot\|_{\otimes_{\alpha}}$.* 2. ii)

$\otimes_{\alpha}$* is injective (projective) if and only if $\|\cdot\|_{\otimes_{\alpha}}$ is injective (resp. projective).*

Proof.

By Propositions 3.6 and 3.7, to prove that $\varPsi$ is a bijection it is only necessary to verify that, whenever $\beta\left(\cdot\right)=\left\|\cdot\right\|_{\otimes_{\alpha}},$ then $\otimes_{\alpha}=\otimes_{\beta}.$ To that end, consider an $l$-tuple $P_{i}\subseteq\mathbb{E}_{i},$ $i=1,...,l,$ of [math]-symmetric convex bodies. Then, applying (3.3) in the first equality and (3.2) in the third equality, we have

[TABLE]

In that case, if $\left\|\cdot\right\|_{\alpha}$ is a tensor norm on finite dimensional normed spaces and $\otimes_{\beta}=\otimes_{\left\|\cdot\right\|_{\alpha}},$ then $\left\|\cdot\right\|_{\otimes_{\beta}}=\left\|\cdot\right\|_{\alpha}.$

To prove $i)$, let $\otimes_{\alpha}$ be a tensor product of [math]-symmetric convex bodies and let $\beta\left(\cdot\right):=\left\|\cdot\right\|_{\otimes_{\alpha}}$. To see that $\beta^{\prime}\left(\cdot\right)=\left\|\cdot\right\|_{\otimes_{\alpha^{\prime}}}$, we have to check that given $l$ finite dimensional normed spaces $M_{i}$,

[TABLE]

To that end, let us fix a scalar product $\left\langle\cdot,\cdot\right\rangle_{i}$ on each $M_{i}$ (we checked in Proposition 3.6 the independence on the selected scalar products).

Let $T_{i}:M_{i}\rightarrow M_{i}^{*}$ be defined as $T_{i}(x):=\left\langle\cdot,x\right\rangle_{i}.$ Then, $T_{i}\left(B_{M_{i}}^{\circ}\right)=B_{M_{i}^{*}},$ and $T_{1}\otimes\cdots\otimes T_{l}:\otimes_{i=1}^{l}M_{i}\rightarrow\otimes_{i=1}^{l}M_{i}^{*}$ is the map sending $w\mapsto\left\langle\cdot,w\right\rangle_{H}.$ Thus, by Proposition 2.7, we have:

[TABLE]

Then, for every $\varphi\in\otimes_{i=1}^{l}M_{i}^{*}$, it holds that $g_{B_{M_{1}^{*}}\otimes_{\alpha}\cdots\otimes_{\alpha}B_{M_{l}^{*}}}\left(\varphi\right)\leq 1$ if and only if $T_{1}^{-1}\otimes\cdots\otimes T_{l}^{-1}\left(\varphi\right)\in B_{M_{1}}^{\circ}\otimes_{\alpha}\cdots\otimes_{\alpha}B_{M_{l}}^{\circ}.$ Since $\left\langle z,T_{1}^{-1}\otimes\cdots\otimes T_{l}^{-1}\left(\varphi\right)\right\rangle_{H}=\varphi\left(z\right),$ we obtain

[TABLE]

We have already proved that $\varPsi$ preserves duality. With this and the bijectivity of $\varPsi$, it follows that $\varPsi^{-1}$ preserves duality, too.

To prove $ii)$ let us suppose that $\otimes_{\alpha}$ is injective. Consider any finite dimensional normed spaces $M_{i}$ and any fixed scalar product on $M_{i}$, $i=1,...,l$, $\left\langle\cdot,\cdot\right\rangle_{i}$.

We consider, on every subspace $N_{i}\subseteq M_{i}$, the scalar product dermined by the restriction of $\left\langle\cdot,\cdot\right\rangle_{i}$ to $M_{i}.$ The injectivity of $\otimes_{\alpha}$ implies

[TABLE]

Since $B_{M_{i}}\cap N_{i}=B_{N_{i}}$, we get that

[TABLE]

This already shows that $\left\|\cdot\right\|_{\otimes_{\alpha}}$ is injective. Since all the steps of the proof are reversible, it follows that the reciprocal is also true.

The projective case follows using $i)$, namely, that the bijection $\varPsi$ preserves duality, along with the dual relation between injective and projective tensor products (Proposition 2.10 and [24, Proposition 7.5]). ∎

Given a tensor norm $\alpha$ on normed spaces, its injective associate tensor norm $/\alpha\setminus$ is the biggest injective tensor norm $/\alpha\setminus\leq\alpha$. Similarly, its projective associate tensor norm $\setminus\alpha/$ is the smallest projective tensor norm $\setminus\alpha/\geq\alpha.$ See [6, 24] for a deeper discussion about these tensor norms. With this notation, we have:

Proposition 3.9**.**

Let $\otimes_{\alpha}$ be a tensor product of [math]-symmetric convex bodies and let $\left\|\cdot\right\|_{\otimes_{\alpha}}$ be its corresponding tensor norm. Then, the tensor norms on finite dimensions that correspond (under the bijection of Theorem 3.8) to the products $\otimes_{\alpha^{inj}},$ $\otimes_{\alpha^{proj}}$ are the injective and projective associated to $\left\|\cdot\right\|_{\otimes_{\alpha}}$ respectively.

Proof.

From (ii) of Theorem 3.8, it follows that $\left\|\cdot\right\|_{\otimes_{\alpha^{inj}}}$ is an injective tensor norm such that $\left\|\cdot\right\|_{\otimes_{\alpha^{inj}}}\leq\left\|\cdot\right\|_{\otimes_{\alpha}}.$ Thus, by the definition of $\otimes_{\alpha^{inj}}$ and Theorem 3.8, it has to be the biggest one below $\left\|\cdot\right\|_{\otimes_{\alpha}}.$ The result for the projective associate tensor norm and $\left\|\cdot\right\|_{\otimes_{\alpha^{proj}}}$ follows from the previous one, the definition of $\otimes_{\alpha^{proj}}$ and the duality exposed in (i) of Theorem 3.8. ∎

3.4. Explicit representation of the biggest injective and the smallest projective tensor products.

Every separable Banach space can be isometrically embedded into $\ell_{\infty}$ and is a quotient of $\ell_{1}$ ([14, pp. 19] and [14, pp. 17]). With these properties, we will obtain explicit representations of $\otimes_{\pi^{inj}}$ and $\otimes_{\epsilon^{proj}}$, respectively.

Lemma 3.10**.**

For each $i=1,\ldots,l$, the following statements hold:

1. (1)

Let $j_{i}:(\mathbb{E}_{i},g_{P_{i}})\hookrightarrow\ell_{\infty}$ and $k_{i}:(\mathbb{E}_{i},g_{P_{i}})\hookrightarrow\ell_{\infty}$ be isometric embeddings. Then,

[TABLE] 2. (2)

Let $q_{i}:\ell_{1}\rightarrow(\mathbb{E}_{i},g_{P_{i}})$ and $h_{i}:\ell_{1}\rightarrow(\mathbb{E}_{i},g_{P_{i}})$ be quotient maps. Then,

[TABLE]

Proof.

(1). We will use the injectivity of the space $\ell_{\infty}$ ([6, Ex. 1.7]) to extend linear mappings defined on subspaces, to linear mappings preserving the norms. The isometries $j_{i}\circ(k_{i})^{-1}:k_{i}(\mathbb{E}_{i})\rightarrow j_{i}(\mathbb{E}_{i})\subset\ell_{\infty}$ can, thus, be extended to norm-one mappings $\mu_{i}:\ell_{\infty}\rightarrow\ell_{\infty}$. The restriction of $\otimes_{i=1}^{n}\mu_{i}:{\ell_{\infty}\hat{\otimes}_{\pi}\cdots\hat{\otimes}_{\pi}\ell_{\infty}}\rightarrow{\ell_{\infty}\hat{\otimes}_{\pi}\cdots\hat{\otimes}_{\pi}\ell_{\infty}}$ satisfies

[TABLE]

The same argument used with the isometries $k_{i}\circ(j_{i})^{-1}$ says that this contention is, indeed, an equality:

[TABLE]

Hence, applying $(j_{1}\otimes\cdots\otimes j_{l})^{-1}$ to the last equality, we obtain the desired result.

(2). This part follows from the lifting property of $\ell_{1}$ ([6, 3.12]). Indeed, let $\rho>0$ and $\tilde{q}_{i}:\ell_{1}\rightarrow\ell_{1}$ be such that $q_{i}=h_{i}\circ\tilde{q}_{i}$ and $\|\tilde{q}_{i}\|\leq(1+\rho)\|q_{i}\|.$ Then, since each $q_{i}$ has norm one, we have that $\tilde{q}_{1}\otimes\cdots\otimes\tilde{q}_{i}(B_{\ell_{1}\hat{\otimes}_{\epsilon}\cdots\hat{\otimes}_{\epsilon}\ell_{1}})\subseteq(1+\rho)^{l}(B_{\ell_{1}\hat{\otimes}_{\epsilon}\cdots\hat{\otimes}_{\epsilon}\ell_{1}}).$ Applying $h_{1}\otimes\cdots\otimes h_{l}$ to the previous inclusion, we have:

[TABLE]

Thus, $q_{1}\otimes\cdots\otimes q_{l}(B_{\ell_{1}\hat{\otimes}_{\epsilon}\cdots\hat{\otimes}_{\epsilon}\ell_{1}})\subseteq h_{1}\otimes\cdots\otimes h_{l}(B_{\ell_{1}\hat{\otimes}_{\epsilon}\cdots\hat{\otimes}_{\epsilon}\ell_{1}}).$

The lifting property used for the quotient maps $h_{i}$ shows the other inclusion. This yields to the desired equality. ∎

Theorem 3.11**.**

For $i=1,...,l$, let $P_{i}\subset\mathbb{E}_{i}$ be [math]-symmetric convex bodies.

1. (1)

For any isometric embeddings $j_{i}:(\mathbb{E}_{i},g_{P_{i}})\hookrightarrow\ell_{\infty},$

[TABLE] 2. (2)

For any quotient mappings $q_{i}:\ell_{1}\rightarrow(\mathbb{E}_{i},g_{P_{i}}),$

[TABLE]

Proof.

(1). Since $\otimes_{\pi}$ corresponds, under the bijection given in Theorem 3.8, to the projective norm $\pi(\cdot),$ then, by Proposition 3.9, $\otimes_{\pi^{inj}}$ corresponds to the biggest injective tensor norm $/\pi\setminus.$ Hence, for each tuple $P_{i}\subset\mathbb{E}_{i},$ $i=1,\ldots,l,$ $P_{1}\otimes_{\pi^{inj}}\cdots\otimes_{\pi^{inj}}P_{l}=B_{\otimes_{/\pi\setminus,i=1}^{l}\left(\mathbb{E}_{i},g_{P_{i}}\right)}.$

If $j_{i}:\left(\mathbb{E}_{i},g_{P_{i}}\right)\hookrightarrow\ell_{\infty}$ are are isometric emmbedings as in [6, 20.7], then $/\pi\setminus(u)=\pi(j_{1}\otimes\cdots\otimes j_{l}(u)),$ for every $u\in\otimes_{i=1}^{l}\left(\mathbb{E}_{i},g_{P_{i}}\right).$ So,

[TABLE]

By Lemma 3.10, the latter does not depend on the election of the embeddings $j_{i}$.

To prove $(2),$ we use that every separable Banach space is a quotient of $\ell_{1}.$ Also, by [6, 20.6], the smallest projective tensor norm $\setminus\epsilon/$ is such that $B_{\otimes_{\setminus\epsilon/}\left(\mathbb{E}_{i},g_{P_{i}}\right)}=(q_{1}\otimes\cdots\otimes q_{l})\left(B_{\ell_{1}\hat{\otimes}_{\epsilon}\cdots\hat{\otimes}_{\epsilon}\ell_{1}}\right),$ where $q_{i}:\ell_{1}\rightarrow(\mathbb{E}_{i},g_{P_{i}})$ are the quotient mappings of [6, 20.6]. Thus, by applying Proposition 3.9 and 3.10, we obtain the desired result. ∎

Corollary 3.12**.**

For $d_{i}\in\mathbb{N}$, $i=1,\ldots,l$,

[TABLE]

[TABLE]

Proof.

Let us denote by $j_{i}:\ell_{\infty}^{d_{i}}\hookrightarrow\ell_{\infty}$ the inclusion mappings into the first $d_{i}$ coordinates and $\Pi_{i}:\ell_{\infty}\rightarrow\ell_{\infty}^{d_{i}}$ the projection mappings. Since $\pi$ is a projective tensor norm, we have that

[TABLE]

is a norm one projection. This, along with Theorem 3.11 implies that

[TABLE]

Hence, since $\Pi_{i}\circ j_{i}$ is the identity on $\mathbb{R}^{d_{i}},$ we have:

[TABLE]

The statement concerning the biggest projective tensor product of $B_{1}^{d_{i}}$ follows by duality. ∎

The arguments given for proving Theorem 3.11 can be adapted to the case where the embeddings are not necessarily isometric. The precise statement is as follows:

Proposition 3.13**.**

For $i=1,\ldots l$ and $d_{i}\in\mathbb{N}$, let $j_{i}:(\mathbb{E}_{i},g_{P_{i}})\hookrightarrow\ell_{\infty}^{d_{i}}$ be embeddings with $\|(j_{i})\|=1$, $\|(j_{i})^{-1}\|\leq M_{i}$. Then

[TABLE]

Furthermore,

[TABLE]

The second part in the last proposition follows from Corollary 3.12, the uniform property and the first part of the proposition.

3.5. Relations with some classes of [math]-symmetric convex bodies

Ellipsoids are fundamental tools in the study of both convex bodies and finite dimensional Banach spaces, as can be seen in [1, 13, 27]. Below we establish the relation between tensor products of convex bodies and John and Löwner ellipsoids. Recall that given a [math]-symmetric convex body $P\subset\mathbb{E},$ its Löwner ellipsoid $L\ddot{o}w(P)$ and John ellipsoid $John(P)$ are the ellipsoids of minimal (resp. maximal) volume containing (resp. contained in) $P.$ We suggest [27, Chapter 3] for a deeper discussion of this matter.

In the following $\otimes_{2}$ denotes the Hilbert tensor product of ellipsoids as defined in Section 2.2 or [10, Section 4]. Since $\otimes_{2}$ is not defined on the class of [math]-symmetric convex bodies, it is not a tensor product of [math]-symmetric convex bodies. However, it is not difficult to show that when restricted to the class of ellipsoids, $\otimes_{2}$ satisfies both properties in Definition 3.1.

Proposition 3.14**.**

Let $\otimes_{\alpha}$ be a tensor product of [math]-symmetric convex bodies and let $P_{i}\subset\mathbb{E}_{i},$ $i=1,\ldots,l$ be any tuple of [math]-symmetric convex bodies.

1. (1)

If $\otimes_{\alpha}\subset\otimes_{2}$ on the class of ellipsoids, then $L\ddot{o}w(P_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}P_{l})=L\ddot{o}w(P_{1})\otimes_{2}\cdots\otimes_{2}L\ddot{o}w(P_{l}).$ 2. (2)

If $\otimes_{2}\subset\otimes_{\alpha}$ on the class of ellipsoids, then $John(P_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}P_{l})=John(P_{1})\otimes_{2}\cdots\otimes_{2}John(P_{l}).$

Proof.

We will prove the result for the Löwner ellipsoid. The proof for the John ellipsoid follows from the duality between Löwner and John ellipsoids, and Proposition 2.10.

Suppose that $\otimes_{\alpha}$ satisfies the inclusion in (1) and let us denote $d_{i}$ the dimension of $\mathbb{E}_{i}$. By [27, Theorem 15.4], there exist $m_{i}$, $x^{i}_{k_{i}}\in\mathbb{E}_{i},$ and positive real numbers $c^{i}_{k_{i}},$ for $k_{i}=1,\ldots,m_{i},$ s.t. $g_{P_{i}}(x_{k_{i}}^{i})=g_{P^{\circ}_{i}}(x_{k_{i}}^{i})=g_{L\ddot{o}w(P_{i})}(x_{k_{i}}^{i})=1,$ $I_{\mathbb{E}_{i}}=\sum_{k_{i}=1}^{m_{i}}c^{i}_{k_{i}}\langle\cdot,x_{k_{i}}^{i}\rangle_{L\ddot{o}w(P_{i})}x^{i}_{k_{i}}$ and $\sum_{k_{i}=1}^{m_{i}}c^{i}_{k}=d_{i}.$ From this and (1) in Definition 3.1, it follows that $I_{\otimes_{i=1}^{l}\mathbb{E}_{i}}$ has a representation as in [27, Theorem 15.4] with the vectors $x_{k_{1}}^{1}\otimes\cdots\otimes x_{k_{l}}^{l},$ the scalars $c^{1}_{k_{1}}\cdots c^{l}_{k_{l}}$ and the scalar product associated to $L\ddot{o}w(P_{1})\otimes_{2}\cdots\otimes_{2}L\ddot{o}w(P_{l}).$ Furthermore, since $\otimes_{\alpha}\subset\otimes_{2},$ then $P_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}P_{l}\subset L\ddot{o}w(P_{1})\otimes_{\alpha}\cdots\otimes_{\alpha}L\ddot{o}w(P_{l})\subset L\ddot{o}w(P_{1})\otimes_{2}\cdots\otimes_{2}L\ddot{o}w(P_{l}).$ Hence, by the uniqueness of the Löwner ellipsoid, $L\ddot{o}w(P_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}P_{l})=L\ddot{o}w(P_{1})\otimes_{2}\cdots\otimes_{2}L\ddot{o}w(P_{l}).$ ∎

Observe that $(1)$ in Proposition 3.14 always holds for $\otimes_{\alpha}=\otimes_{\pi}$. This case was already proved in [2, Lemma 1] . It is worth to notice that also $\otimes_{2}\subset\otimes_{\epsilon}$ is always fulfilled. Thus, Proposition 3.14 proves that the John ellipsoid of the injective tensor product of [math]-symmetric convex bodies is the Hilbert tensor product of the John ellipsoids.

Remark 3.15*.*

The inclusions in Proposition 3.14 can not be dropped. To prove this, observe that $John(B_{2}^{d_{1}}\otimes_{\pi}\cdots\otimes_{\pi}B_{2}^{d_{l}})\subset B_{2}^{d_{1}}\otimes_{\pi}\cdots\otimes_{\pi}B_{2}^{d_{l}}$ is not the Hilbert tensor product of $B_{2}^{d_{i}},$ $i=1,\ldots,l.$

The class of [math]-symmetric convex bodies with enough symmetries (the unit balls $B_{p}^{d}$, for example) is a well known class of convex sets used, among others, to study Banach-Mazur distances, see [27, $\mathsection$16]. Given a [math]-symmetric convex body $P\subset\mathbb{E}$, the set of symmetries of $P$ is denoted by $Iso(P).$ It consists of the linear maps $w:\mathbb{E}\rightarrow\mathbb{E}$ such that $wP=P.$ Recall that a [math]-symmetric convex body $P$ has enough symmetries if the only linear maps $T:\mathbb{E}\rightarrow\mathbb{E}$ that conmute with each $w\in Iso(P)$ are the scalar multiples of the identity map $I_{\mathbb{E}}$.

Proposition 3.16**.**

Let $\otimes_{\alpha}$ be a tensor product of [math]-symmetric convex bodies. If $P_{i}\subset\mathbb{E}_{i}$, $i=1,\ldots,l,$ are [math]-symmetric convex bodies with enough symmetries then $P_{1}\otimes_{\alpha}\cdots\otimes_{\alpha}P_{l}$ has enough symmetries too.

Proof.

The result is proved using an inductive argument. Note that the case of one factor is trivial. We briefly expose the proof for the case of three factors. The case of two factors and the general one, follow analogously. Induction is used to prove that the corresponding equalities (3.4) imply the existence of the scalar $\lambda$ below. In [3, Excercise 4.27] one finds the case of two factors for the product $\otimes_{\pi}$.

To that end, observe that by Proposition 2.7, $w_{1}\otimes w_{2}\otimes w_{3}\in Iso(P_{1}\otimes_{\alpha}P_{2}\otimes_{\alpha}P_{3})$ for every $w_{i}\in Iso(P_{i}).$ Let $S:\mathbb{E}_{1}\otimes\mathbb{E}_{2}\otimes\mathbb{E}_{3}\rightarrow\mathbb{E}_{1}\otimes\mathbb{E}_{2}\otimes\mathbb{E}_{3}$ be a linear map that conmutes with the elements of $Iso(P_{1}\otimes_{\alpha}P_{2}\otimes_{\alpha}P_{3}).$

For each $a^{2},b^{2}\in\mathbb{E}_{2}$ and $a^{3},b^{3}\in\mathbb{E}_{3},$ let $S_{a^{2}b^{2}a^{3}b^{3}}$ be the linear map on $\mathbb{E}_{1}$ defined via duality as $\langle S_{a^{2}b^{2}a^{3}b^{3}}(x),x^{\prime}\rangle_{\mathbb{E}_{1}}=\langle S(x\otimes a^{2}\otimes a^{3}),x^{\prime}\otimes b^{2}\otimes b^{3}\rangle_{H}.$ It is not difficult to see that $S_{a^{2}b^{2}a^{3}b^{3}}$ conmutes with each $w_{1}\in Iso(P_{1}).$ Hence, $S_{a^{2}b^{2}a^{3}b^{3}}=\lambda_{a^{2}b^{2}a^{3}b^{3}}I_{\mathbb{E}_{1}}.$ In a similar way, one can define $S_{a^{1}b^{1}a^{3}b^{3}}:\mathbb{E}_{2}\rightarrow\mathbb{E}_{2},$ $S_{a^{1}b^{1}a^{2}b^{2}}:\mathbb{E}_{3}\rightarrow\mathbb{E}_{3}$ for every $a^{1}b^{1}\in\mathbb{E}_{1}.$ Clearly, $S_{a^{1}b^{1}a^{3}b^{3}}=\lambda_{a^{1}b^{1}a^{3}b^{3}}I_{\mathbb{E}_{2}},$ $S_{a^{1}b^{1}a^{2}b^{2}}=\lambda_{a^{1}b^{1}a^{2}b^{2}}I_{\mathbb{E}_{3}}$ also hold. Thus by evaluating on $a^{i},b^{i}\in\mathbb{E}_{i},$ $i=1,2,3,$ we have:

[TABLE]

for every $a^{i},b^{i}\in\mathbb{E}_{i}$. The latter implies that, $\lambda_{a^{2}b^{2}a^{3}b^{3}}=\lambda\langle a^{1},b^{1}\rangle_{\mathbb{E}_{1}},$ $\lambda_{a^{1}b^{1}a^{3}b^{3}}=\lambda\langle a^{2},b^{2}\rangle_{\mathbb{E}_{2}}$ and $\lambda_{a^{1}b^{1}a^{2}b^{2}}=\lambda\langle a^{3},b^{3}\rangle_{\mathbb{E}_{3}}$ for some $\lambda.$ Hence, $S=\lambda I_{\mathbb{E}_{1}\otimes\mathbb{E}_{2}\otimes\mathbb{E}_{3}}.$ ∎

4. Order 2 tensor products of [math]-symmetric convex bodies and ideals of linear operators

In the case of products of order two, the theory of tensor products of Banach spaces is closely related to the theory of operators ideals. This fact can be found in the monographs [6, 7, 24, 27].

Following the usual terminology, a Banach operator ideal consists of an assignment to each pair of Banach spaces $X,Y$ of a vector space $\mathcal{A}\left(X,Y\right)$ of bounded linear operators from $X$ to $Y,$ together with a norm $A$ on this space, with the following properties:

1. (1)

$\mathcal{A}\left(X,Y\right)$ is a linear subspace of $\mathcal{L}\left(X,Y\right)$ which contains the finite rank operators. Furthermore, for every $\varphi\in X^{*}$ and $y\in Y$, $A\left(\varphi\left(\cdot\right)y\right)=\left\|\varphi\right\|\left\|y\right\|$. 2. (2)

The ideal property: if $S\in\mathcal{A}\left(X_{0},Y_{0}\right),T\in\mathcal{L}\left(X,X_{0}\right)$ and $R\in\mathcal{L}\left(Y_{0},Y\right)$ then $RST\in\mathcal{A}\left(X,Y\right)$ and $A\left(RST\right)\leq\left\|R\right\|A\left(S\right)\left\|T\right\|.$ 3. (3)

$\left(\mathcal{A}\left(X,Y\right),A\right)$ is a Banach space.

If we only consider finite dimensional normed spaces, then $\mathcal{A}(M,N)=\mathcal{L}(M,N)$ with the norm $A$. Thus, in this context, we say that $A$ is an ideal norm. For a complete treatment of Banach operator ideals see [6, 21], and for the case of finite dimensions see [27].

Let $u=\sum_{i=1}^{n}x_{i}\otimes y_{i}$ be an element of the tensor product $M\otimes N$ of finite dimensional normed spaces. By $T_{u}$ we denote the linear map:

[TABLE]

Conversely, let $T=\sum_{i=1}^{n}x_{i}^{*}\left(\cdot\right)y_{i}$ be a linear map from $M$ to $N.$ By $u_{T}$ we denote the vector $\sum_{i=1}^{n}x_{i}^{*}\otimes y_{i}$ that belongs to $M^{*}\otimes N$. The mappings $u_{T}\mapsto T_{u}$ and $T_{u}\mapsto u_{T}$ establish a natural bijection between tensor norms on finite dimensional spaces and ideal norms (see 17.1 and 17.2 of [6]). Then, along with Theorem 3.8, we have:

Corollary 4.1**.**

Given a tensor product of order $2$ of [math]-symmetric convex bodies $\otimes_{\alpha}$, for every pair of finite dimensional normed spaces $M,N$, define:

[TABLE]

Then $A$ is an ideal norm.

Indeed, we have that the mappings in the following diagram, which are as in Theorem 3.8 and Corollary 4.1, are bijections that respect duality, injectivity and projectivity:

[TABLE]

4.1. The Hilbertian tensor product of [math]-symmetric convex bodies

Let $(\mathcal{L}_{H},\|\cdot\|_{H})$ be the operator ideal of bounded operators which factor through a Hilbert space. If $T\in\mathcal{L}_{H}(X;Y)$, then $\|T\|_{H}=inf\{\|R\|\,\|S\|\}$ where the infimum runs over all possible factorizations. Here we follow the notation in [7] and [24]. In [22] this ideal and the norm are denoted $(\Gamma_{2},\gamma_{2})$. The tensor norm $\omega_{2}$ associated to this ideal is known as the Hilbertian tensor norm. It is profusely studied in [6], [7], [24]. It is an injective tensor norm. A fundamental result in the theory of tensor norms involves $\omega_{2},$ it is the so called Grothendieck‘s Theorem. It is formulated, in terms of tensor norms, as:

[TABLE]

where $K_{G}$ is Grothendieck’s constant and $n,m\in\mathbb{N}.$ This form of Grothendieck’s theorem appears as [6, 14.4]. We suggest the reader the monograph [23] on the subject.

According to (4.1), let us define $\otimes_{\omega_{2}}$ as the Hilbertian tensor product of [math]-symmetric convex bodies associated to the tensor norm $\omega_{2}$ and the ideal $\|\cdot\|_{H}$. Thus, by means of Theorem 3.8, Grothendieck‘s theorem can be written in terms of [math]-symmetric convex bodies as:

[TABLE]

By Theorem 3.8, $\otimes_{\omega_{2}}$ is an injective tensor product. Combining these results, it also holds:

Proposition 4.2**.**

Let $P_{i}\subset\mathbb{E}_{i}$, $i=1,2$ be [math]-symmetric convex bodies. Then

[TABLE]

Proof.

The result in terms of tensor norms can be found in [24, Theorem 7.29]. Now, it is enough to apply the transition to the [math]-symmetric convex bodies setting, that Theorem 3.8 and Proposition 3.9 provide. ∎

These results give rise to an explicit representation (up to a constant) of the tensor product $\otimes_{\omega_{2}}$. In the following corollary it is worth noticing that, by Dvoretsky‘s Theorem ([1, Theorem 5.1.2]), one may consider $M_{i}\leq 1+\epsilon$ for an arbitrary $\epsilon>0$, at the price of growing the dimensions $d_{i}$:

Corollary 4.3**.**

Let $P_{i}\subset\mathbb{E}_{i}$, $i=1,2,$ be [math]-symmetric convex bodies and let $j_{i}:(\mathbb{E}_{i},g_{P_{i}})\hookrightarrow\ell_{\infty}^{d_{i}}$ be embeddings with $\|j_{i}\|=1,\,\|(j_{i})^{-1}\|\leq M_{i}$, as in Proposition 3.13. Then

[TABLE]

In the case where $P_{1}:=\mathcal{E}_{1}$ is an ellipsoid (the case $P_{2}:=\mathcal{E}_{2}$ is analogue)

[TABLE]

Proof.

The first part follows from Proposition 4.2 and Proposition 3.13. To prove the assertion on the tensor product of ellipsoids, we are using the notation ${\simeq}$ to mean the norm of the isomorphisms, which is as above.

To prove the assertion on ellipsoids, let us denote $F:=(\mathbb{E}_{2},g_{P_{2}})$. Observe that $(\mathcal{L}(H_{1}^{*},F),\|\cdot\|)$ is isometric to $(\mathcal{L}(H_{1}^{*},F),\|\cdot\|_{H})$ since, trivially, every bounded operator factorizes through a Hilbert space. By Theorem 3.8 and (4.1), this means that ${\mathcal{E}}_{1}\otimes_{\epsilon}P_{2}={\mathcal{E}}_{1}\otimes_{\omega_{2}}P_{2}$. The isomorphic expression as a section of the projective tensor product of $\ell_{\infty}$-spaces follows from the previous one. ∎

The dual tensor product $\otimes_{\omega_{2}^{\prime}}$ is well defined, according to Definition 2.9. Also, it is a projective tensor product. By Theorem 3.8, it corresponds to the dual tensor norm $\omega_{2}^{\prime}.$ Even more, from Proposition 4.2, it follows that $\otimes_{\omega_{2}^{\prime}}$ is equivalent to the biggest projective tensor product ${\otimes_{\epsilon^{proj}}}$. That is:

Corollary 4.4**.**

For every pair of [math]-symmetric convex bodies $P_{i}\subset\mathbb{E}_{i},$ $i=1,2,$ the following holds:

[TABLE]

As a consequence of the last corollary and Proposition 3.13, both of the statements in Corollary 4.3 have dual expresions for the tensor products $\otimes_{\omega_{2}^{\prime}}$ and $\otimes_{\epsilon}.$

Let $P_{i}\subset\mathbb{E}_{i}$ be [math]-symmetric convex bodies and $q_{i}:\ell_{1}^{d_{i}}\hookrightarrow(\mathbb{E}_{i},g_{P_{i}}),$ $i=1,2,$ be surjective linear maps such that $q_{i}(B_{1}^{d_{i}})\subset P_{i}\subset M_{i}q_{i}(B_{1}^{d_{i}})$, for $M_{i}>0,$ $i=1,2$. Then,

[TABLE]

In the case where $P_{1}:=\mathcal{E}_{1}$ is an ellipsoid, it holds:

[TABLE]

It is worth to notice that, in the context of tensor norms, Corollary 4.4 corresponds to [24, Corollary 7.30]. For a detail treatment of the norm $\omega_{2}^{\prime},$ we suggest the reader [24, Section 7.4].