Dual and Generalized Dual Cones in Banach Spaces

TL;DR

This paper explores dual cones, faces, and projections in Banach spaces, highlighting differences from Hilbert spaces and providing illustrative examples to deepen understanding.

Contribution

It introduces and analyzes dual cones, faces, and visions in Banach spaces, extending concepts from Hilbert space theory and examining their properties.

Findings

Dual cones lose key properties when moving from Hilbert to Banach spaces

Relations between faces, visions, and projections are established in Banach spaces

Illustrative examples demonstrate the theoretical results

Abstract

The primary objective of this paper is to propose and analyze the notion of dual cones associated with the metric projection and generalized projection in Banach spaces. We show that the dual cones, related to the metric projection and generalized metric projection, lose many important properties in transitioning from Hilbert spaces to Banach spaces. We also propose and analyze the notions of faces and visions in Banach spaces and relate them to the metric projection and generalized projection. We provide many illustrative examples to give insight into the given results.

Equations

$$B(r)$$

$$B^0(r)$$

$$S(r)$$

$$⟨Jx,x⟩=\Vert Jx{\Vert }_{{\text{X}}^{\ast }}\Vert x{\Vert }_{\text{X}}=\Vert x{\Vert }_{\text{X}}^2=\Vert Jx{\Vert }_{{\text{X}}^{\ast }}^2,\text{for any}x\in X.$$

$$\Vert z{\Vert }_3=\sqrt[3]{|z_1{|}^3+|z_2{|}^3+|z_3{|}^3}.$$

$$\Vert \psi {\Vert }_{\frac{3}{2}}={\left(|{\psi }_1{|}^{\frac{3}{2}}+|{\psi }_2{|}^{\frac{3}{2}}+|{\psi }_3{|}^{\frac{3}{2}}\right)}^{\frac{2}{3}}.$$

$$Jz=\left(\frac{|z_1{|}^2\text{sign}(z_1)}{\Vert z{\Vert }_3},\frac{|z_2{|}^2\text{sign}(z_2)}{\Vert z{\Vert }_3},\frac{|z_3{|}^2\text{sign}(z_3)}{\Vert z{\Vert }_3}\right).$$

$$J^{\ast }\psi =\left(\frac{|{\psi }_1{|}^{\frac{3}{2}-1}\text{sign}({\psi }_1)}{{\left(\Vert \psi {\Vert }_{\frac{3}{2}}\right)}^{\frac{3}{2}-2}},\frac{|{\psi }_2{|}^{\frac{3}{2}-1}\text{sign}({\psi }_2)}{{\left(\Vert \psi {\Vert }_{\frac{3}{2}}\right)}^{\frac{3}{2}-2}},\frac{|{\psi }_3{|}^{\frac{3}{2}-1}\text{sign}({\psi }_3)}{{\left(\Vert \psi {\Vert }_{\frac{3}{2}}\right)}^{\frac{3}{2}-2}}\right).$$

$$V(\psi ,x)=\Vert \psi {\Vert }_{X^{\ast }}^2-2⟨\psi ,x⟩+\Vert x{\Vert }_X^2,\text{for any}\psi \in X^{\ast },x\in X.$$

$$\Vert x-P_{C}x{\Vert }_X\le \Vert x-z{\Vert }_X,\text{for all}z\in C.$$

$$V(\psi ,{\pi }_C\psi )=\underset{y\in C}{\inf }V(\psi ,y),\text{for any}\psi \in X^{\ast }.$$

$$u=P_C(x)\iff ⟨J_X(x-u),u-z⟩\ge 0,\text{for all}z\in C.$$

$$y={\pi }_C(\psi ),\text{if and only if},⟨\psi -J_{X}y,y-z⟩\ge 0,\text{for all}z\in C.$$

$$P_C^{-1}(y)=\{u\in X:P_C(u)=y\}.$$

$$K^{\perp }=\{x\in H:⟨x-v,v-z⟩\ge 0,\text{for all}z\in K\}.$$

$$⟨x,P_{K}x⟩=\Vert P_{K}x{\Vert }^2,\text{for all}x\in H.$$

$$K_P^{\perp }=\{x\in X:⟨J(x-v),v-z⟩\ge 0,\text{for all}z\in K\}.$$

$$⟨J(v+t(x-v)-v),v-z⟩=⟨J(t(x-v)),v-z⟩=t⟨J(x-v),v-z⟩\ge 0,\text{for all}z\in K,$$

$$P_K(x)=v\iff ⟨J(x-v),v-z⟩\ge 0,\text{for all}z\in K.$$

$$J(x-\theta )=\left(\frac{9}{\Vert x{\Vert }_3},\frac{-4}{\Vert x{\Vert }_3},\frac{-1}{\Vert x{\Vert }_3}\right)=\left(\frac{9}{\sqrt[3]{36}},\frac{-4}{\sqrt[3]{36}},\frac{-1}{\sqrt[3]{36}}\right).$$

$$⟨J(x-\theta ),\theta -\alpha u⟩=⟨\left(\frac{9}{\sqrt[3]{36}},\frac{-4}{\sqrt[3]{36}},\frac{-1}{\sqrt[3]{36}}\right),-\alpha (-25,-37,-77)⟩=0,\text{for all}\alpha u\in K,\alpha \in [0,\infty ),$$

$$⟨J(y-\theta ),\theta -\alpha u⟩=⟨\left(\frac{1}{\sqrt[3]{36}},\frac{-9}{\sqrt[3]{36}},\frac{4}{\sqrt[3]{36}}\right),-\alpha (-25,-37,-77)⟩=0,\text{for all}\alpha u\in K,\alpha \in [0,\infty ),$$

$$J(h-\theta )=\frac{7\sqrt[3]{4}}{6}(1,-1,0).$$

$$⟨J(h-\theta ),\theta -\alpha u⟩=-14\sqrt[3]{4}\alpha <0,\text{for every}\alpha u\in K,\alpha \in (0,\infty ),$$

$$⟨J(\alpha u-\theta ),\theta -x⟩=\alpha ⟨Ju,-x⟩<0,$$

$$⟨Jw,P_{K}w⟩\ne \Vert P_{K}w{\Vert }_X^2,\text{for}w\in X.$$

$$⟨J(tx-t{P}_{K}x),t{P}_{K}x-z⟩=t^2⟨J(x-P_{K}x),P_{K}x-t^{-1}z⟩\ge 0,\text{for all}z\in K,$$

$$⟨J(w-u),u-\alpha u⟩=(1-\alpha )⟨\left(\frac{9}{\sqrt[3]{36}},\frac{-4}{\sqrt[3]{36}},\frac{-1}{\sqrt[3]{36}}\right),(-25,-37,-77)⟩=0,\text{for any}\alpha u\in K,\alpha \in [0,\infty ).$$

$$\Vert w{\Vert }_X=\sqrt[3]{28^3+35^3+76^3}<\Vert u{\Vert }_X=\sqrt[3]{25^3+37^3+77^3},$$

$$|⟨Jw,P_{K}w⟩|=|⟨Jw,u⟩|\le \Vert Jw{\Vert }_{X^{\ast }}\Vert u{\Vert }_X=\Vert w{\Vert }_X\Vert u{\Vert }_X<\Vert u{\Vert }_X^2,$$

$${\pi }_C^{-1}(v)=\{\psi \in X^{\ast }:{\pi }_C(\psi )=v\}.$$

Full text

1 Introduction

Dual cones, induced by the metric projections, have a simple structure and valuable properties in the setting of Hilbert spaces. The derivations of such properties heavily exploit the underlying Hilbertian structure. The Hilbertian structure also equips the metric projection with attractive features, see Zarantonello (Zar71, , Lemma 1.5). However, during the last three decades, many important studies of metric projection have been conducted in Banach spaces. This development is partly motivated by the real-world applications of metric projection in optimization, approximation theory, inverse problems, variational inequalities, image processing, neural networks, machine learning, and others. For an overview of these details and some of the related developments, see BalGol12 ; BalMarTei21 ; Bau03 ; BorDruChe17 ; Bou15 ; Bro13 ; BroDeu72 ; Bui02 ; Bur21 ; CheGol59 ; ChiLi05 ; Den01 ; DeuLam80 ; DutShuTho17 ; GJKS21 ; Ind14 ; FitPhe82 ; KonLiuLiWu22 ; KroPin13 ; Li04 ; Li04a ; LiZhaMa08 ; Nak22 ; Osh70 ; Pen05 ; PenRat98 ; QiuWan22 ; Ric16 ; Sha16 ; ShaZha17 ; ZhaZhoLiu19 , and the cited references.

The primary objective of this research is to propose and analyze the notion of dual cones associated with the metric projection in Banach spaces. We note that the shortcomings of the metric projection in Banach spaces have resulted in important extensions, namely, the generalized projection and the generalized metric projection, which enjoy better properties in a Banach space framework, see Alb93 ; Alb96 ; KhaLiRwi22 ; Li04a ; Li05 . We show that the dual cones, related to the metric projection and generalized metric projection, lose many properties in transitioning from Hilbert spaces to Banach spaces. We also propose and analyze the notions of faces and visions in Banach spaces and relate them to the metric projection and generalized projection. Illustrative examples are given.

The contents of this paper are organized into five sections. After a brief introduction in Section 1, we recall some background material Section 2. Section 3 studies dual cones related to the metric projection where as the dual cones related to the generalized projection are studied in Section 4. various notions of projections and give new results concerning normalized duality mapping. Section 5 studies the faces and visions in Banach spaces.

2 Preliminaries

Let $X$ be a real Banach space with norm $\|\cdot\|_{X}$, let $X^{*}$ be the topological dual of $X$ with norm $\|\cdot\|_{X^{*}}$, and let $\langle\cdot,\cdot\rangle$ be the duality pairing between $X^{*}$ and $X$. We will denote the null elements in $X$ and $X^{*}$ by $\theta$ and $\theta^{*}.$ Moreover, the closed and convex hull of a set $M\in X$ is denoted by $\overline{\text{co(M)}}.$ Given a Banach space $X$, for $r>0$, we denote the closed ball, open ball and sphere with radius $r$ and center $\theta$ by

[TABLE]

For details on the notions recalled in this section, see Tak00 .

Given a uniformly convex and uniformly smooth Banach space $X$ with dual space $X^{*}$, the normalized duality map $J:X\to X^{*}$ is a single-valued mapping defined by

[TABLE]

In a uniformly convex and uniformly smooth Banach space $X$, the normalized map $J_{X}:X\to X^{*}$ is one-to-one, onto, continuous and homogeneous. Furthermore, the normalized duality mapping $J^{*}:X^{*}\to X$ is the inverse of $J$, that is $J^{*}J=I_{X}$ and $JJ^{*}=I_{X^{*}}$, where $I_{X}$ and $I_{X}^{*}$ are the identity maps in $X$ and $X^{*}$. On the other hand, in a general Banach space $X$ with dual $X^{*}$, the normalized duality mapping $J:X\to 2^{X^{*}}$ is a set-valued mapping with nonempty valued. In particular, if $X^{*}$ is strictly convex, then $J:X\to X^{*}$ is a single-valued mapping. See Tak00 .

The following example will be repeatedly used in this work.

Example 2.1**.**

Let $X=\mathds{R}^{3}$ be equipped with the $3$-norm $\|\cdot\|_{3}$ defined for any $z=(z_{1},z_{2},z_{3})\in X,$ by

[TABLE]

Then $(X,\|\cdot\|_{3})$ is a uniformly convex and uniformly smooth Banach space (and is not a Hilbert space). The dual space of $(X,\|\cdot\|_{3})$ is $(X^{*},\|\cdot\|_{\frac{3}{2}})$ so that for any $\psi=(\psi_{1},\psi_{2},\psi_{2})$, we have

[TABLE]

The normalized duality mappings $J$ and $J^{*}$ satisfy the following conditions. For any $z=(z_{1},z_{2},z_{3})\in X$ with $z\neq\theta$, we have

[TABLE]

Moreover, for any $\psi=(\psi_{1},\psi_{2},\psi_{3})\in X^{*}$ with $\psi\neq\theta$, we have

[TABLE]

Let $X$ be a uniformly convex and uniformly smooth Banach space and let $C$ be a nonempty, closed, and convex subset of $X$. We define a Lyapunov function $V:X^{*}\times X\to\mathds{R}$ by the formula:

[TABLE]

We shall now recall useful notions of projections in Banach spaces.

Definition 2.2**.**

Let $X$ be a uniformly convex and uniformly smooth Banach space, let $X^{*}$ be the dual of $X$, and let $C$ be a nonempty, closed, and convex subset of $X$.

The metric projection $P_{C}:X\to C$ is a single-valued defined by

[TABLE]

The generalized projection $\pi_{C}:X^{*}\to C$ is a single-value map that satisfies

[TABLE]

The following result collects some of the basic properties of the metric projection defined above.

Proposition 2.3**.**

Let $X$ be a uniformly convex and uniformly smooth Banach space and let $C$ be a nonempty, closed, and convex subset of $X$.

The metric projection $P_{C}:X\to C$ is a continuous map that enjoys the following variational characterization:

[TABLE]

The generalized projection $\pi_{C}:X^{*}\to C$ enjoys the following variational characterization: For any $\psi\in X^{*}$ and $y\in C,$

[TABLE]

We will also need the following notions. Given a Banach space $X$, for any $u,v\in X$ with $u\neq v$, we write

($a$)

$[v,u]=\{tv+(1-t)u:\ 0\leq t\leq 1\}.$

($b$)

$[v,u\lceil=\{tv+(1-t)u:\ 0\leq t<\infty\}.$

($c$)

$\rceil u,v\lceil=\{tv+(1-t)u:\ -\infty<t<\infty\}.$

The set $[v,u]$ is a closed segment with end points $u$ and $v$. The set $[v,u\lceil$ is a closed ray in $X$ with end point $v$ with direction $u-v$, which is a closed convex cone with vertex at $v$ and is a special class of cones in $X$. The set $\rceil u,v\lceil$ is a line in $X$ passing through points $v$ and $u$.

We conclude this section by recalling the following result (see KhaLi23 ):

Theorem 2.4**.**

Let $X$ be a uniformly convex and uniformly smooth Banach space and let $C$ a nonempty, closed, and convex subset of $X$. For any $y\in C,$ let $x\in X\backslash C$ be such that $y=P_{C}{x}$. We define the inverse image of $y$ under the metric projection $P_{C}:X\to C$ by

[TABLE]

Then $P_{C}^{-1}(y)$ is a closed cone with vertex at $y$ in $X$. However, $P_{C}^{-1}(y)$ is not convex, in general.

3 Dual Cones for the Metric Projection

A cone $K$ in a vector space is said to pointed if $K$ has vertex at $\theta$, $K\cap(-K)=\{\theta\}$, and $K\neq\theta.$

Let $H$ be a Hilbert space, and let $K$ be a cone in $H$ with vertex at $v$. We define the dual cone of $K$ in $H$ with respect to the metric projection $P_{K}$ by

[TABLE]

The dual cone has the following properties in Hilbert spaces (see Zarantonello Zar71 ):

(1)

$K^{\perp}$ is a closed and convex cone in $H$ with vertex at $v$.

(2)

$K^{\perp\perp}=\overline{\text{co}K}$.

(3)

If $K$ is a closed and convex cone, then $K^{\perp}$ and $K$ are dual cones of each other.

(4)

If $K$ is a closed, convex and pointed cone, then $P_{K}$ is positive homogeneous and

[TABLE]

In this following, we extend the concept of a dual cone from Hilbert spaces to uniformly convex and uniformly smooth Banach spaces and derive their valuable properties. We will show that the properties (3) and (4) given above do not hold, in general, in Banach spaces.

Definition 3.1**.**

Let $X$ be a Banach space, let the dual $X^{*}$ of $X$ be strictly convex, and let $K$ be a cone in $X$ with vertex $v$. We define the dual cone with respect to the metric projection $P$ by

[TABLE]

The following result shows that $K_{P}^{\perp}$ is a cone in $X$, and $K_{P}^{\perp}$ and $K$ have the same vertex $v$.

Theorem 3.2**.**

Let $X$ be a Banach space, let the dual $X^{*}$ of $X$ be strictly convex, and let $K$ be a cone in $X$ with vertex $v$. Then, the following statements hold:

($a$)

$K_{P}^{\perp}$* is a cone with vertex at $\theta$ in $X$.*

($b$)

If $X$ is uniformly convex and uniformly smooth, then $K_{P}^{\perp}$ is closed.

($c$)

If $X$ is uniformly convex and uniformly smooth and $K$ is closed and convex, then $K_{P}^{\perp}=P_{K}^{-1}(v).$

($d$)

$K_{P}^{\perp}$* is not convex.*

($e$)

$K\not\subseteq(K_{P}^{\perp})_{P}^{\perp}.$**

($f$)

$K$* and $K_{P}^{\perp}$ are not dual of each other.*

Proof.

($a$) For an arbitrary $x\in K_{P}^{\perp}$ with $x\neq v$, and for any $t>0$, by the homogeneity property of the normalized duality mapping $J$, we have

[TABLE]

implying that $v+t(x-v)\in K_{P}^{\perp},$ for all $t>0.$ Thus, $K_{P}^{\perp}$ is a cone in $X$ with vertex at $v$.

($b$) Under the additional hypothesis on $X$, $J$ is continuous, which proves that $K_{P}^{\perp}$ is closed.

($c$) By the basic variational principle of $P_{K}$, for any given $x\in X,$ we have

[TABLE]

Since (7) coincides with (6), we deduce that $K_{P}^{\perp}=P_{K}^{-1}(v).$

(d) We construct a counterexample to show that $K_{P}^{\perp}$ is not convex. Take $X=\mathds{R}_{3}$ given in Example 2.1. Let $u=(-25.-37,-77)$, and $K=[\theta,u\lceil=\{\alpha u:\ 0\leq\alpha<\infty\}.$ Take $x=(3,-2,-1)$ and $y=(1,-3,2).$ Then $\|x\|_{3}=\|y\|_{3}=\sqrt[3]{36}.$ By using (2.1), we have

[TABLE]

Next, we compute

[TABLE]

which implies that $x\in K_{P}^{\perp}$. Analogously, $J(y-\theta)=\left(\frac{1}{\sqrt[3]{36}},\frac{-9}{\sqrt[3]{36}},\frac{4}{\sqrt[3]{36}}\right),$ and hence

[TABLE]

which proves that $y\in K_{P}^{\perp}.$ For $h=\frac{2}{3}x+\frac{1}{3}y$, we have $h-\theta=h=\left(\frac{7}{3},-\frac{7}{3},0\right)$, yielding

[TABLE]

We now compute

[TABLE]

which proves that $h\neq K_{P}^{\perp}$. Thus, $K_{P}^{\perp}$ is not convex.

($e$) Since $K_{P}^{\perp}$ is a closed cone with vertex at $\theta$, $(K_{P}^{\perp})_{P}^{\perp}$ is a closed cone with vertex at $\theta$. We will use the counterexample from (d). Recall that $u=(-25,-37,-77)$ and $K=[\theta,u\lceil$. We showed that $x=(3,-2,-1)\in K_{P}^{\perp}$. Then,

[TABLE]

which implies that $\alpha u\notin(K_{P}^{\perp})_{P}^{\perp}$, for any $\alpha u\in K$ with $\alpha\in(0,\infty)$, which prove ($e$). Finally, ($f$) follows from (e). ∎

Proposition 3.3**.**

Let $X$ be a uniformly convex and uniformly smooth Banach space and let $K$ be a closed, convex, and pointed cone in $X$. Then $P_{K}$ is positive homogeneous. In general,

[TABLE]

Proof.

For any $t>0$, since $K$ is a closed, convex and pointed cone in $X$, for any $x\in X$, we have

[TABLE]

and by appealing to the basic variational property of $P_{K}$, this implies that $P_{K}(tx)=tP_{K}x.$

We next construct a counterexample to prove (8). Let $x=\mathds{R}^{3}$ be as in Example 2.1. Let $u=(-25,-37,-77)$ and $K=[\theta,u\lceil$. We take a point $w=(-28,-35,-76)$. Then $w-u=(3,-2,-1).$ By the proof of Theorem 3.2, we have

[TABLE]

By the basic variational principle, we have $P_{K}(w)=u.$ Next we calculate,

[TABLE]

which implies that

[TABLE]

which verifies (8). The proof is complete. ∎

4 Generalized dual cones with respect to the generalized projection $\pi$

We now study the generalized dual cone of $K$ for the generalized projection $\pi$. We first recall some properties of the inverse image of the vertex of a cone by the generalized projection $\pi_{C}$ in $X^{*}$.

Given a uniformly convex and uniformly smooth Banach space $X$ with dual space $X^{*}$ and a cone $K$ with vertex at $v$, we recall that

[TABLE]

Theorem 4.1**.**

Let $X$ be a uniformly convex and uniformly smooth Banach space with dual $X^{*}$ and let $K$ be a closed and convex cone in $X$ with vertex at $v$. Then,

(a)

$\pi_{K}^{-1}(v)$* is a closed and convex cone in $X^{*}$ with vertex at $Jv$.*

(b)

$\pi^{-1}_{\pi_{K}^{-1}(v)}(Jv)=K.$**

Proof.

(a) See Theorem 2.4. (b) For a fixed $z\in K$, we have

[TABLE]

which, taking into account the identity $J^{*}=J^{-1}$, implies that

[TABLE]

By the basic variational principle for $\pi_{\pi_{K}^{-1}(v)}$, we obtain that $z\in\pi_{\pi_{K}^{-1}(v)}^{-1}(Jv),$ for all $z\in K,$ proving

[TABLE]

For the converse, for any $z\in\pi_{\pi_{K}^{-1}(v)}^{-1}(Jv)$, we have $\pi_{\pi_{K}^{-1}(v)}(z)=Jv$. Appealing to the variational principle for $\pi_{\pi^{-1}_{K}(v)}$ once again, we have

[TABLE]

and hence

[TABLE]

We recall that for $\ell\in X^{*}$, we have

[TABLE]

For the given $z\in\pi_{\pi_{K}^{-1}(v)}^{-1}(Jv)$, let $y=z-P_{K}z+v.$ Then, for any $x\in K,$ we define

[TABLE]

Since $K$ is a closed and convex cone with vertex $v$, for all $x\in K,$ we have that $w\in K$. By the variational principle for $P_{K}$, we have

[TABLE]

which implies that $P_{K}y=v$, that is, $y\in P_{K}^{-1}(v)$. Now, let

[TABLE]

Then $\psi\in X^{*}$. By (12), we have

[TABLE]

By (11) and (14), we have $\psi\in\pi_{K}^{-1}(v)$. Since $z\in\pi_{\pi_{K}^{-1}(v)}^{-1}(Jv)$, by the variational principle, we have

[TABLE]

that is,

[TABLE]

which, due to the containment $\psi\in\pi_{K}^{-1}(v)$, implies that

[TABLE]

Then, using (13), we have $\langle J(y-v),v-z\rangle\geq 0.$ Then,

[TABLE]

which due to $\langle J(z-P_{K}z),P_{K}z-v\rangle\geq 0$ implies that $\|z-P_{K}z\|^{2}=0,$ that is, $z=P_{K}z\in K.$ Since $z$ is arbitrary in $\pi_{\pi_{K}^{-1}(v)}^{-1}(Jv)$, we obtain $\pi_{\pi_{K}^{-1}(v)}^{-1}(Jv)\subseteq K$. This, in view of (10), completes the proof.∎

Definition 4.2**.**

Let $X$ be a uniformly convex and uniformly smooth Banach space with dual $X^{*}$, and let $K$ be a cone in $X$ with vertex at $v$. We define the generalized dual cone of $K$ in $X^{*}$ with respect to the generalized projection $\pi$ by

[TABLE]

Theorem 4.3**.**

Let $X$ be a uniformly convex and uniformly smooth Banach space with dual $X^{*}$, and let $K$ be a cone in $X$ with vertex at $v$. Then the following statements hold:

(a)

$K_{\pi}^{\perp}=\pi_{K}^{-1}(v)$.

(b)

$K_{\pi}^{\perp}$* is a closed and convex cone with vertex at $Jv$ in $X^{*}$.*

(c)

$K$* and $K_{\pi}^{\perp}$ are generalized dual of each other: $K=(K_{\pi}^{\perp})_{\pi}^{\perp}.$*

Proof.

(a). By the basic variational principle for $\pi_{K}$, for any $\psi\in X^{*}$ and $v\in K$, we have

[TABLE]

which, due to the definition of $K_{\pi}^{\perp}$ and (9) at once implies (a). (b) Since $\pi_{K}^{-1}(v)$ is a closed and convex cone with vertex at $Jv$ in $X^{*}$, (b) follows at once (a). Finally, (c) follows from (a) and Theorem 4.1. ∎

Corollary 4.4**.**

Let $X$ be a uniformly convex and uniformly smooth Banach space, and let $C$ and $K$ be closed and convex cones in $X$ with a common vertex at $v$ satisfying $C\cap K\neq\{v\}$. Then,

(a)

$C\subseteq K\quad\Leftrightarrow\quad C_{\pi}^{\perp}\supseteq K_{\pi}^{\perp}.$**

(b)

$(C\cap K)_{\pi}^{\perp}=\overline{\text{co}(C_{\pi}^{\perp}\cup K_{\pi}^{\perp})}.$**

Proof.

(a) The proof of $C\subseteq K\ \Rightarrow\ C_{\pi}^{\perp}\supseteq K_{\pi}^{\perp}$ is evident. The converse follows from part (c) of Theorem 4.3.

(b) It follows at once that $C\cap K$ is a closed and convex cone in $X$ with vertex $v$. By (a), the inclusion $C\cap K\subseteq C$ implies that $(C\cap K)_{\pi}^{\perp}\supseteq C_{\pi}^{\perp}$ and the inclusion $C\cap K\subseteq K$ implies that $(C\cap K)_{\pi}^{\perp}\supseteq K_{\pi}^{\perp}$, and hence $(C\cap K)_{\pi}^{\perp}\supseteq C_{\pi}^{\perp}\cap C_{\pi}^{\perp}.$ However, since $(C\cap K)_{\pi}^{\perp}$ is a closed and convex cone with vertex at $Jv$, it follows that

[TABLE]

By (c) of Theorem 4.3 and (a), we have

[TABLE]

On the other hand, from $C_{\pi}^{\perp}\subseteq\overline{\text{co}(C_{\pi}^{\perp}\cup K_{\pi}^{\perp})}$ and $K_{\pi}^{\perp}\subseteq\overline{\text{co}(C_{\pi}^{\perp}\cup K_{\pi}^{\perp})}$, we have $C=(C_{\pi}^{\perp})_{\pi}^{\perp}\supseteq\overline{\text{co}(C_{\pi}^{\perp}\cup K_{\pi}^{\perp})}$ and $K=(K_{\pi}^{\perp})_{\pi}^{\perp}\supseteq\overline{\text{co}(C_{\pi}^{\perp}\cup K_{\pi}^{\perp})}$. Thus, by (16), we have

[TABLE]

which proves the desired identity. Since $(C\cap K)_{\pi}^{\perp}$ and $\overline{\text{co}(C_{\pi}^{\perp}\cup K_{\pi}^{\perp})}$ are both closed and convex cones with vertex at $Jv$, we have the result by using Theorem 4.3. ∎

The following result can be proved in an analogous fashion:

Corollary 4.5**.**

Let $X$ be a uniformly convex and uniformly smooth Banach space, and let $\{K_{\lambda}:\ \lambda\in\Lambda\}$ be a set of closed and convex cones with a common vertex at $v$ such that $\cap_{\lambda\in\Lambda}K_{\lambda}\neq\{v\}.$ Then

[TABLE]

where $\Lambda$ is an arbitrary given index set.

5 Faces and visions in Banach spaces

5.1 Faces in Banach spaces

Definition 5.1**.**

Let $X$ be a Banach space with dual $X^{*}$ and let $C$ be a nonempty, closed, and convex subset of $X$. For any $\psi\in X^{*},$ we define the face of $\psi$ on $C$ by

[TABLE]

Remark 5.2**.**

It is evident from the above definition that for any $\psi\in X^{*},$ the set $\mathcal{F}_{C}(\psi)$ is either empty or a closed and convex subset of $C$. Moreover, $\mathcal{F}_{C}(\theta^{*})=C.$

Before proceeding any further, we gather a few examples to illustrate the above notion.

Example 5.3**.**

Let $X=\mathds{R}^{3}$ be as in Example 2.1. We take $u=(25,37,77)$ and let $C=[\theta,u\lceil=\{tu\in\mathds{R}^{3}:\ t\geq 0\}.$

(a)

Let $\psi=(-9,4,1)\in(\mathds{R}^{3})^{*}$. Then $\mathcal{F}_{C}(\psi)=C.$

(b)

Let $\psi=(\psi_{1},\psi_{2},\psi_{3})\in(\mathds{R}^{3})^{*}\ \text{with}\ \psi_{i}\leq 0,\ \text{for}\ i=1,2,3\ \text{and}\ \psi_{1}+\psi_{1}+\psi_{3}<0.$ Then $\mathcal{F}_{C}(\psi)=\{\theta\}.$

(c)

Let $\psi=(\psi_{1},\psi_{2},\psi_{3})\in(\mathds{R}^{3})^{*}\ \text{with}\ \psi_{i}\geq 0,\ \text{for}\ i=1,2,3\ \text{and}\ \psi_{1}+\psi_{1}+\psi_{3}>0.$ Then $\mathcal{F}_{C}(\psi)=\emptyset.$

Proof.

(a). For $\psi=(-9,4,1)\in(\mathds{R}^{3})^{*}$, we have

[TABLE]

which implies that $tu\in\mathcal{F}_{C}(\psi)$ for all $tu\in C$. Parts (b) and (c) can be proved analogously. ∎

Example 5.4**.**

Let $(S,\mathcal{A},\mu)$ be a measure space with $\mu(S)\geq 1.$ For any $p\in[1,\infty),$ let $X=L_{p}(S)$ be the real Banach space of real functions defined on $S$ with norm $\|\cdot\|_{p}$. For any given $M>0$, let

[TABLE]

Then $C$ is a nonempty, closed, and convex subset in $L_{p}(S)$. For any $A\in\mathcal{A}$ with $1\leq\mu(A)<\infty$, let $1_{A}$ denote the characteristic function of $A$, which satisfies $1_{A}\in L_{q}(S)^{*}=L_{p}(S)$, where $p,q\in[1,\infty]$ are such that $\frac{1}{p}+\frac{1}{q}=1.$ Then $\mathcal{F}_{C}(1_{A})$ is a nonempty, closed, and convex subset of $C$ such that

[TABLE]

Proof.

For any $g\in C,$ if $\int_{A}g(s)d\mu(s)=M$, then

[TABLE]

For any $f\in C,$ we have

[TABLE]

Thus, (18) and (19) imply that

[TABLE]

For the converse, we define $h$ on $S$ by

[TABLE]

By $1\leq\mu(A)<\infty$, we have

[TABLE]

which implies that $h\in C$. By $1_{A}\in L_{p}(S)^{*}$, we have

[TABLE]

By the above equation, it follows that for any $g\in\mathcal{F}_{C}(1_{A})\subseteq C$, we must have

[TABLE]

It follows that $\langle 1_{A},g\rangle=M$, that is, $\int_{A}g(s)d\mu(s)=M.$ This implies that

[TABLE]

By combining (20) and (22), we get (17). By (21) and (20), we have $h\in\mathcal{F}_{C}(1_{A})$, which shows that $\mathcal{F}_{C}(1_{A})\neq\emptyset.$ This prove the claim. ∎

Example 5.5**.**

For any $p$ with $1\leq p<\infty$, let $X=\ell_{p}$ be the real Banach space of real sequences with norm $\|\cdot\|_{p}$. For any given $M>0$, let

[TABLE]

Then $C$ is a nonempty, closed and convex subset in $\ell_{p}.$ For any positive integers $m$ and $n$ with $n\geq 1.$ We define $\ell_{m}^{n}\in(\ell_{p})^{*}=\ell_{q}$ by

[TABLE]

Then, $\mathcal{F}_{C}(\ell_{m}^{n})$ is a nonempty, closed, and convex subset of $C$ such that

[TABLE]

Proof.

We only need to show that $\mathcal{F}_{C}(\ell_{m}^{n})$ is nonempty. For this, we take $z=\{z_{i}\}\in\ell_{p}$ as follows:

[TABLE]

Then, it is easy to verify that $z\in C$ and $z\in\mathcal{F}_{C}(\ell_{m}^{n})$. The rest of the arguments are similar to the ones used in Example 5.4. ∎

Lemma 5.6**.**

Let $X$ be a reflexive Banach space with dual space $X^{*}$ and let $C$ be a closed, convex and bounded set in $X$. Then for each $\psi\in X^{*}$, $\mathcal{F}_{C}(\psi)$ is nonempty, closed, and convex subset of $C$.

Proof.

Since $C$ is weakly compact, for any $\psi\in X^{*}$, the function $\langle\psi,\cdot\rangle$ attains its maximum value on $C$. That is, there is $y\in C$ such that $\langle\psi,y\rangle=\max_{x\in C}\langle\psi,x\rangle$. This implies that $y\in\mathcal{F}_{C}(\psi).$ The set $\mathcal{F}_{C}(\psi)$ is clearly, closed and convex. ∎

Theorem 5.7**.**

Let $X$ be a reflexive Banach space with dual $X^{*}$ and let $C$ be a nonempty, closed, and convex set in $X$. Then

(a)

For any $u\in X,$

[TABLE]

(b)

For any $\psi\in X^{*},$

[TABLE]

Proof.

(a) For an arbitrary $z\in C,$ by the basic variational principle for $P_{C},$ we have

[TABLE]

This proves the first equality in (a). To prove the second inequality, for any $z\in C$, by the basic variational principle of $\pi_{C}$, we have

[TABLE]

which proves the second equality in (a).

(b) For any $\psi\in X^{*},$ $J^{*}\psi\in X$ by substituting $J^{*}\psi$ for $u\in X$ in (a) and noticing $JJ^{*}\psi=\psi$, (b) follows at once. ∎

The conclusion of Theorem 5.7 can be described by the form of variational inequalities.

Corollary 5.8**.**

Let $X$ be a uniformly convex and uniformly smooth Banach space wth dual $X^{*}$ and let $C$ be a nonempty, closed, and convex set in $X$. Then

(a)

For any $u\in X,$ a point $y\in C$ is a solution of the variational inequality

[TABLE]

if and only if, $y$ is a solution to one of the following projection equations:

[TABLE]

(b)

For any $\psi\in X^{*},$ a point $y\in C$ is a solution of the variational inequality

[TABLE]

if and only if, $y$ is a solution to one of the following projection equations:

[TABLE]

5.2 Visions in Banach spaces

Definition 5.9**.**

Let $X$ be a Banach space with dual $X^{*}$ and let $C\subset X$ be nonempty, closed, and convex.

(a)

We define the vision $\mathcal{F}_{C}^{-1}(y)$ in $X^{*}$ of a point $y\in C$ with respect to the background $C$ by

[TABLE]

(b)

We define the vision $\mathcal{F}_{C}^{-2}(y)$ in $X$ of a point $y\in C$ with respect to the background $C$ by

[TABLE]

Lemma 5.10**.**

Let $X$ be a uniformly convex and uniformly smooth Banach space with dual $X^{*}$, and let $C$ be a nonempty, closed, and convex subset in $X$. Then, for any $y\in C,$ we have

[TABLE]

Proof.

Since in a uniformly convex and uniformly smooth Banach space $X$, $J$ and $J^{*}$ are both one-to-one and onto mapping such that $J^{*}J=I_{X}$ and $JJ^{*}=I_{X^{*}},$ the conclusions are evident. ∎

Proposition 5.11**.**

Let $X$ be a Banach space with dual $X^{*}$ and let $C\subset X$ be nonempty, closed, and convex. Then, for any $y\in C,$ we have

(a)

$\theta^{*}\in\mathcal{F}_{C}^{-1}(y)$* and $\mathcal{F}_{C}^{-1}(y)\neq\emptyset.$*

(b)

If $\{\theta^{*}\}\subsetneqq\mathcal{F}_{C}^{-1}(y)$, then $\mathcal{F}_{C}^{-1}(y)$ is a closed and convex cone with vertex at $\theta^{*}.$

Proof.

Since (a) is evident, we only prove (b). For any $\psi\in\mathcal{F}_{C}^{-1}(y)$ and $t\geq 0$, we have

[TABLE]

which implies that $t\psi\in\mathcal{F}_{C}^{-1}(y)$, and hence $\mathcal{F}_{C}^{-1}(Y)$ is a cone with vertex at $\theta^{*}$ in $X^{*}.$

For any $\psi,\phi\in\mathcal{F}_{C}^{-1}(y)$ and for any $t\in[0,1]$ by $y\in C$, we have

[TABLE]

which implies that

[TABLE]

and hence $t\psi+(1-t)\phi\in\mathcal{F}_{C}^{-1}(y)$, proving the desired convexity. To prove that $\mathcal{F}_{C}^{-1}(y)$ is closed in $X^{*}$. Let $\{\psi_{n}\}\subseteq\mathcal{F}_{C}^{-1}(y)$ and $\psi\in X^{*}$ be such that $\psi_{n}\to\psi$ in $X^{*}$ as $n\to\infty.$ This implies that

[TABLE]

which proves that $\psi\in\mathcal{F}_{C}^{-1}(y)$, and hence $\mathcal{F}_{C}^{-1}(y)$ is closed in $X^{*}$. ∎

Proposition 5.12**.**

Let $X$ be a Banach space with dual space $X^{*}$ and let $C$ be a nonempty, closed, and convex subset in $X$. Then for any $y\in C$, we have

(a)

$\theta in\mathcal{F}_{C}^{-2}(y)$* and $\mathcal{F}_{C}^{-2}(y)\neq\emptyset.$*

(b)

If $\mathcal{F}_{C}^{-2}\varsupsetneq\{\theta\}$, then $\mathcal{F}_{C}^{-2}(y)$ is a closed cone with vertex at $\theta$ in $X$. In general $\mathcal{F}_{C}^{-2}(y)$ is not convex.

Proof.

We only prove that $\mathcal{F}_{C}^{-2}(y)$ is not convex. Let $X=\mathds{R}^{3}$ be as in Example 2.1. Let $y=(25,37,77)$ and define $C=[\theta,y]=\{\alpha y:\ 0\leq\alpha\leq 1\}$. We take $x=(3,-2,-1)$ and $z=(1,-3,2)$. Then $\|x\|_{3}=\|z\|_{3}=\sqrt[3]{36}.$ As before, we compute

[TABLE]

Therefore, $x\in\mathcal{F}_{C}^{-2}(y)$. Analogously, we have $z\in\mathcal{F}_{C}^{-2}(y)$. We take $h=\frac{2}{3}x+\frac{1}{3}z=\left(\frac{7}{3},-\frac{7}{3},0\right).$ Proceeding as before, we have

[TABLE]

proving that $h\notin\mathcal{F}_{C}^{-2}(y).$ Thus, $\mathcal{F}_{C}^{-2}(y)$ is not convex which proves the assertion. ∎

Definition 5.13**.**

Let $X$ be a Banach space with dual $X^{*}$ and let $C$ be a nonempty, closed, and convex set in $X$. For any $y\in C,$ we define

(a)

If $\mathcal{F}_{C}^{-1}(y)=\{\theta^{*}\}$, then $y$ is called an internal point of $C$.

(b)

If $\mathcal{F}_{C}^{-1}(y)\supsetneqq\{\theta^{*}\}$, then $y$ is called a cuticle point of $C$.

The collection of all internal points of $C$ is denoted by $\mathcal{J}(C)$and the collection of all cuticle points of $C$ is denoted by $\mathcal{C}(C)$

As a direct consequence of Proposition 5.11, we have the following result.

Corollary 5.14**.**

Let $X$ be a Banach space with dual $X^{*}$ and let $C$ be a nonempty, closed, and convex set in $X$. Then $C=\{\mathcal{J}(C),\mathcal{C}(C)\}$ is a partition of $C$. More precisely, we have $C=\mathcal{J}(C)\cup\mathcal{C}(C).$

Corollary 5.15**.**

Let $X$ be a uniformly convex and uniformly smooth Banach space with dual $X^{*}$ and let $C$ be a nonempty, closed, and convex set in $X$. For any $y\in C,$ we have

(a)

$y\in\mathcal{J}(C)$* if and only if for $\psi\in X^{*}$, $y=\pi_{C}(\psi+Jy)$ implies that $\psi=\theta^{*}.$*

(b)

$y\in\mathcal{C}(C)$* if and only if there is $\theta^{*}\neq\psi\in X^{*}$ such that $y=\pi_{C}(\psi+Jy)$.*

An analogue of the above result can be given by using the metric projection $P_{C}$.

Corollary 5.16**.**

Let $X$ be a uniformly convex and uniformly smooth Banach space with dual $X^{*}$ and let $C$ be a nonempty, closed, and convex set in $X$. For any $y\in C,$ we have

(a)

$y\in\mathcal{J}(C)$* if and only if for $\psi\in X^{*}$, $y=P_{C}(J^{*}\psi+y)$ implies that $\psi=\theta^{*}.$*

(b)

$y\in\mathcal{C}(C)$* if and only if there is $\theta^{*}\neq\psi\in X^{*}$ such that $y=P_{C}(J^{*}\psi+y)$.*

Next we give some examples to demonstrate the concepts of $\mathcal{J}(C)$ and $\mathcal{C}(C).$

Corollary 5.17**.**

Let $X$ be Banach space with dual $X^{*}$ and let $C$ be a proper closed subspace of $X$. Then:

(a)

$\mathcal{J}(C)=\emptyset.$**

(a)

$\mathcal{C}(C)=C.$**

Proof.

Since $C$ is a proper closed subspace of $X$, by the Hahn-Banach space theorem, there is $\psi\in X^{*}$ with $\|\psi\|_{X^{*}}=1$ such that $\langle\psi,x\rangle=0$ for all $x\in C.$ This implies that $\psi\in\mathcal{F}_{C}^{-1}(y)$ for all $y\in C.$ Since $\psi\neq\theta^{*}$, it follows at once that $y\in\mathcal{C}(C)$, for all $y\in C$. The claim then follows from Corollary 5.14. ∎

In the following result, we use the closed and open balls and the unit sphere, see Section 2.

Proposition 5.18**.**

Let $X$ be Banach space with dual $X^{*}$. For $r>0$, we have

(a)

$\mathcal{J}(B(r))=B^{0}(r).$**

(b)

$\mathcal{C}(B(r))=S(r).$**

(c)

For any $y\in S(r)$, $\mathcal{F}_{B(r)}^{-1}(y)$ is a closed and convex cone with vertex at $\theta^{*}$ and

[TABLE]

(d)

If $X^{*}$ is strictly convex, then for any $y\in S(r)$, we have

[TABLE]

Proof.

(a) We first prove that $\theta\in\mathcal{J}(B(r))$. For any $\psi\in X^{*}$ with $\|\psi\|_{X^{*}}>0$, there is $x\in B(r)$ such that $\langle\psi,x\rangle\neq 0.$ Since for $x\in B(r)$, we also have $-x\in B(r)$, it follows that one of $\langle\psi,x\rangle$ and $\langle\psi,-x\rangle$ is positive. By $\langle\psi,\theta\rangle=0,$ it follows that

[TABLE]

This implies $\mathcal{F}_{B(r)}^{-1}(\theta)=\{\theta^{*}\}$, and therefore $\theta\in\mathcal{J}(B(r)).$

For any $y\in B^{0}(r)$ with $0<\|y\|_{X}<r$, the proof of $y\in\mathcal{J}(B(r))$ is divided into two parts.

Case 1. $\psi\in X^{*}$ with $\|\psi\|_{X^{*}}>0$ satisfying $\langle\psi,y\rangle=0.$ Then, there is $z\in B(r)$ and $-z\in B(r)$ such that $\langle\psi,z\rangle\neq 0.$ It follows that one of $\langle\psi,x\rangle$ and $\langle\psi,-x\rangle$ is positive. Then,

[TABLE]

Case 2. $\psi\in X^{*}$ with $\|\psi\|_{X^{*}}>0$ satisfying $\langle\psi,y\rangle\neq 0.$ By $0<\|y\|_{X}<r$, there are positive numbers $s$ and $t$ with $t>1>s>0$ such that $\|ty\|_{X}<r$. Then, $ty,ts\in B^{0}(r)\subset B(r)$. We have

[TABLE]

By $ty,ts\in B(r)$, we deduce that

[TABLE]

Combining (24) and (25), for any $y\in B^{0}(r)$ with $0<\|y\|_{X}<r$, we have

[TABLE]

which implies that

[TABLE]

which, when combined with the containment $\theta\in\mathcal{J}(B(r))$ proves (a).

(b) For any $y\in S(r)$ and for any $\psi\in Jy\subseteq X^{*}$, we have $\|\psi\|_{X^{*}}=\|y\|_{X}=r$ and $\langle\psi,y\rangle=r^{2}$. Then,

[TABLE]

which implies

[TABLE]

Since $\psi\in X^{*}$ with $\psi\neq\theta$, we have $y\in\mathcal{C}(B(r))$, for any $y\in S(r)$. This, taking into account Remark 5.2 , implies that $\mathcal{C}(B(r))=S(r).$

(c). From (26), we have

[TABLE]

Next we show that for any fixed $y\in S(r),$ we have

[TABLE]

From (26), for any $t\geq 0,$ we have

[TABLE]

which implies that $t\psi\in\mathcal{F}_{B(r)}^{-1}(y)$, for any $t\geq 0$, which proves (28). Therefore,

[TABLE]

On the other hand, for $y\in S(r)$ and for given $\psi\in\mathcal{F}_{B(r)}^{-1}(y)$ with $\psi\neq\theta^{*}$, as in the proof of (28), we can show that

[TABLE]

So, we may assume that $\|\psi\|_{X^{*}}=r$. It follows that $\|J^{*}\psi\|_{X}=r$, which implies $J^{*}\psi\in S(r)$. By $y\in S(r)$, $\psi\in\mathcal{F}_{B(r)}^{-1}(y)$ and $J^{*}\psi\in S(r)$, it follows that

[TABLE]

This implies $\|\psi\|_{X^{*}}=\|y\|_{X}=r$ and $\langle\psi,y\rangle=r^{2}$ Hence $\psi\in Jy.$ We have established,

[TABLE]

implying

[TABLE]

By combining (30) and (32), we complete the proof of (c).

(d) It follows at once from (c) under the additional hypothesis on $X$. ∎

The following result connects generalized dual cones with the notion of visions.

Theorem 5.19**.**

Let $X$ be a uniformly convex and uniformly smooth Banach space and let $K$ be a closed and convex cone in $X$ with vertex at $v$. Then,

[TABLE]

Proof.

By Theorem 4.3, we have $K_{\pi}^{\perp}=\pi_{K}^{-1}(v)$. Thus, we only need to prove the second equality in (33). For any $\psi\in X^{*}$, we have

[TABLE]

and the proof is complete. ∎

Remark 5.20**.**

Equation (33) reexamines the following results:

(i)

$K_{\pi}^{\perp}$* is a closed and convex cone with vertex at $Jv$ in $X^{*}$ (Theorem 4.3).*

(ii)

$\mathcal{F}_{K}^{-1}(v)$* is a closed and convex cone with vertex at $\theta^{*}$ in $X^{*}$ (Proposition 5.12).*