Canonical decomposition of a difference of convex sets

TL;DR

This paper introduces a canonical convex decomposition for the difference of two convex sets in a lattice setting and relates the volume of the pieces to intersection numbers of toric b-divisors, advancing geometric understanding.

Contribution

It provides a canonical decomposition of the difference of convex sets and links the volume of parts to intersection theory of toric b-divisors, a novel geometric interpretation.

Findings

Canonical convex decomposition of the difference of convex sets.

Geometric interpretation of volumes via intersection numbers.

Enhanced understanding of convex set differences in lattice geometry.

Abstract

Let $N$ be a lattice of rank $n$ and let $M = N^{\vee}$ be its dual lattice. In this note we show that given two compact, bounded, full-dimensional convex sets $K_1 \subseteq K_2 \subseteq M_{\R} \coloneqq M \otimes_{\Z} \R$, there is a canonical convex decomposition of the difference $K_2 \setminus K_1$ and we interpret the volume of the pieces geometrically in terms of intersection numbers of toric $b$-divisors.

Equations

$$\Upsilon\left(K_{2}\setminus K_{1}\right)\coloneqq\left\{{\operatorname{convhull}}\left(F_{1},F_{2}\right)\,\big{|}\,F_{1}\overset{{\operatorname{exposed}}}{\leq}K_{1},\,F_{2}\overset{{\operatorname{exposed}}}{\leq}K_{2}\,\text{ and }\,F_{1}\sim F_{2}\right\}$$

$${\mathbf{D}}_2^n-{\mathbf{D}}_1^n=\sum _{F\stackrel{\mathrm{exposed}}{\le }{K}_2}{c}_F,$$

$$c_F$$

$$F=\left\{m\in K\,\big{|}\,\langle v,m\rangle=\min_{m^{\prime}\in K}\langle v,m^{\prime}\rangle\right\}.$$

$${\varphi }_K:N_{\mathbb{R}}⟶\underline{\mathbb{R}}(=\mathbb{R}\cup \{-\infty \})$$

$$v⟼\underset{m\in K}{\inf }⟨m,v⟩.$$

$$f^{\vee }:M_{\mathbb{R}}⟶\underline{\mathbb{R}},$$

$$m⟼\underset{v\in N_{\mathbb{R}}}{\inf }\left(⟨m,v⟩-f(v)\right),$$

$${\iota }_K:M_{\mathbb{R}}⟶\underline{\mathbb{R}}$$

$${\iota }_K(m)=\{\begin{array}{ll}0& \text{ if }m\in K,\\ -\infty & \text{ if }m\notin K.\end{array}$$

$$\partial f(u)\coloneqq\left\{m\in M_{{\mathbb{R}}}\,\big{|}\,\langle m,u-v\rangle\geq f(u)-f(v),\,\forall\,v\in N_{{\mathbb{R}}}\right\},$$

$$\partial f(V):=\bigcup _{u\in V}\partial f(u).$$

$$\mathrm{relint}(\mathrm{dom}(f))\subseteq \mathrm{dom}(\partial f)\subseteq \mathrm{dom}(f).$$

$$\mathrm{relint}(K_f)\subseteq \mathrm{Im}(\partial f)\subseteq K_f.$$

$$C_m:=\partial f^{\vee }(m)\subseteq \mathcal{P}(N_{\mathbb{R}}),$$

$$\mathcal{L}f:{\rm Y}(f)⟶{\rm Y}(f^{\vee })$$

$$C⟼\bigcap _{u\in C}\partial f(u)\left(=\partial f(u_0),\text{ for any }{u}_0\in \mathrm{relint}(C)\right).$$

$${\rm Y}⟶{{\rm Y}}^{\ast }$$

$$C⟼C^{\ast }$$

$${\varphi }_K(a,b)=\{\begin{array}{ll}\frac{ab}{a+b},& \text{ if }a,b\in {\mathbb{R}}_{\ge 0},\\ \min \{0,a,b\},& \text{ otherwise. }\end{array}$$

$${\Sigma }_{K_i}:={\rm Y}({\varphi }_{K_i})$$

$$\mathrm{relint}\left({\sigma }_{F_1}\right)\cap \mathrm{relint}\left({\sigma }_{F_2}\right)\ne \varnothing$$

$$\Upsilon\left(K_{2}\setminus K_{1}\right)\coloneqq\left\{{\operatorname{convhull}}\left(F_{1},F_{2}\right)\,\big{|}\,F_{1}\overset{{\operatorname{exposed}}}{\leq}K_{1},\,F_{2}\overset{{\operatorname{exposed}}}{\leq}K_{2}\,\text{ and }\,F_{1}\sim F_{2}\right\}$$

$$F_{1}=\left\{m\in K_{1}\,\big{|}\,\langle v,m\rangle=\min_{m^{\prime}\in K_{1}}\langle v,m^{\prime}\rangle\right\}\quad\text{ and }\quad F_{2}=\left\{m\in K_{2}\,\big{|}\,\langle v,m\rangle=\min_{m^{\prime}\in K_{2}}\langle v,m^{\prime}\rangle\right\}.$$

$$m={\lambda }_1{m}_1+{\lambda }_2{m}_2$$

$$⟨v,m_1⟩=\underset{m^{\prime }\in K_1}{\min }⟨v,m^{\prime }⟩\ge \underset{m^{\prime }\in K_2}{\min }⟨v,m^{\prime }⟩=⟨v,m_2⟩.$$

$$⟨v,m⟩={\lambda }_1⟨v,m_1⟩+{\lambda }_2⟨v,m_2⟩\le {\lambda }_1⟨v,m_1⟩+{\lambda }_2⟨v,m_1⟩=⟨v,m_1⟩,$$

$$\mathrm{convhull}(F_1,F_2)\cap K_1=F_1,$$

$$K_{2}\setminus K_{1}=\left\{{\operatorname{convhull}}\left(F_{1},F_{2}\right)\,\bigg{|}\,F_{1}\overset{\text{\tiny{exposed}}}{\leq}K_{1},\,F_{2}\overset{\text{\tiny{exposed}}}{\leq}K_{2}\,\text{ and }\,F_{1}\sim F_{2}\right\}.$$

$$\mathrm{convhull}\left(F_1,F_2\right)\cap \mathrm{convhull}\left(F_1^{\prime },F_2^{\prime }\right)$$

Full text

0 Introduction

Convex sets have been widely and successfully used to explore the geometry of an algebraic variety using convex geometrical methods. A well known class of examples comes from the theory of toric varieties, where the combinatorics of a lattice polypte encrypts most of the geometric properties of the corresponding projective toric variety (see [CLS10] and [Ful93]). More generally, Okounkov bodies (in the literature often called Newton–Okounkov bodies) are convex sets which one can attach to an algebraic variety together with some extra geometric data, e.g. a complete flag of subvarieties. These convex sets turn out to encode also important geometric information of the variety (see [Oko96, Oko03] and also [KK12, KK14, KK08] and [LM09] and the references therein).

More recently, generalizing the toric situation, in [Bot17], convex sets are associated to so called toric $b$-divisors, which can be though of as a limit of toric divisors keeping track of birational information. Their degree is defined as a limit. There it is shown that under some positivity assumptions toric b-divisors are integrable and that their degree is given as the volume of a convex set. Moreover, it is shown that the dimension of the space of global sections of a nef toric $b$-divisor is equal to the number of lattice points in this convex set and a Hilbert–Samuel type formula for its asymptotic growth is given. This generalizes classical results for classical toric divisors on toric varieties. Finally, a relation between convex bodies associated to $b$-divisors and Okounkov bodies is established. We remark that the main motivation for studying toric $b$-divisors is to be able to do arithmetic intersection theory on mixed Shimura varieties of non-compact type. Indeed, it turns out that toric $b$-divisors locally encode the singularities of the invariant metric on an automorphic line bundle over a toroidal compactification of a mixed Shimura variety of non-compact type. This note is part of an overall program to develop an arithmetic intersection theory on mixed Shimura varieties of non-compact type via convex geometric methods whose starting point is [Bot17].

In general, of a particular interest is to compute the volume of a convex set. Aside from this being intrinsically a question of great interest, it has applications not only in the above mentioned geometric settings, but also in other mathematical fields such as in convex optimization.

For the rest of this introduction let us fix a lattice $N$ of rank $n$, its dual lattice $M=N^{\vee}$, and two compact, bounded, full-dimensional convex sets $K_{1}\subseteq K_{2}\subseteq M_{{\mathbb{R}}}=M\otimes_{{\mathbb{Z}}}{\mathbb{R}}$. The aim of this note is to show there is a canonical convex decomposition of the difference $K_{2}\setminus K_{1}$ and to interpret geometrically the volume of the pieces in terms of intersection numbers of toric $b$-divisors.

The outline of this note is as follows. In Section 1 we recall the Legendre–Fenchel duality for convex sets. Most of the definitions and statements which we will state regarding this duality can be found in [Roc72]. We also refer to [BPS14, Chapter 2].

In Section 2 we give the canonical convex decomposition of the difference $K_{2}\setminus K_{1}$. We start by defining what it means for two faces $F_{1}$ and $F_{2}$ of $K_{1}$ and $K2$, respectively, to be related, denoted by $F_{1}\sim F_{2}$. Using this relationship, we are able to show the following main result of this section, which is Theorem 2.5 in the text.

Theorem 0.1**.**

Let notations be as above. Then we have that

[TABLE]

is a convex decomposition of the difference $K_{2}\setminus K_{1}$.

In the polyhedral case, the above canonical decomposition gives a polyhedral subdivision of the complement of two polytopes, one contained in the other. This subdivision appears in the literature (e.g. in [GP88]) although it is constructed using the so called pushing method. We haven’t found in the literature the method we used in Theorem 2.5 nor have we found such a canonical decomposition in the non-polyhedral case.

In Section 3, we start by recalling the definition of toric $b$-divisors from [Bot17] and, in the nef case, their connection with convex sets. We then recall the definition of the surface area measure (and a mixed version thereof) associated to a convex set (and to a collection of convex sets) for which our main reference is the survey of Schneider [Sch93]. Finally, we relate this measure to the intersection theory of toric $b$-divisors.

In Section 4, we give a geometric interpretation of the above canonical decomposition in terms of intersection numbers of toric $b$-divisors in the case that $K_{2}$ is polyhedral. The main result of this section is the following, which is Theorem 4.2 in the text.

Theorem 0.2**.**

Let notations be as above and assume that $K_{2}$ is a polytope. Let ${\boldsymbol{D}}_{1}$ and ${\boldsymbol{D}}_{2}$ be the nef toric $b$-divisors on $X_{\Sigma_{K_{2}}}$ corresponding to the convex sets $K_{1}$ and $K_{2}$, respectively, where $\Sigma_{K_{2}}$ denotes the normal fan of $K_{2}$. Then we can express the difference ${\boldsymbol{D}}_{2}^{n}-{\boldsymbol{D}}_{1}^{n}$ of degrees of the nef toric $b$-divisors ${\boldsymbol{D}}_{1}$ and ${\boldsymbol{D}}_{2}$ as a finite sum of correction terms

[TABLE]

where the sum is over all exposed faces of $K_{2}$. The correction terms $c_{F}$ are related to top intersection numbers of toric $b$-divisors by

[TABLE]

where $S(\cdot)$ is the mixed surface area measure associated to a collection of convex sets; and where $\phi_{1}$ and $\phi_{2}$ denote the support functions of the convex sets $K_{1}$ and $K_{2}$, respectivly.

Acknowledgements I would like to thank J. I. Burgos for enlightning discussions and several corrections and comments on earlier drafts of this note.

1 Legendre–Fenchel duality

Throughout this section, $N$ will denote a lattice of rank $n$ and $M=N^{\vee}$ its dual lattice. All convex sets in $M_{{\mathbb{R}}}$ are assumed to be compact and full-dimensional. We start with some definitions.

Definition 1.1**.**

Let $K$ be a convex set in $M_{{\mathbb{R}}}$. A convex subset $F\subseteq K$ is called a face of $K$ if, for every closed line segment $\left[m_{1},m_{2}\right]\subseteq K$ such that ${\operatorname{relint}}\left(\left[m_{1},m_{2}\right]\right)\cap F\neq\emptyset$, the inclusion $\left[m_{1},m_{2}\right]\subseteq F$ holds. A non-empty subset $F\subseteq K$ is called an exposed face of $K$ if there exists a $v\in N_{{\mathbb{R}}}$ such that

[TABLE]

Remark 1.2**.**

Every exposed face is a face. However, not every face is exposed, as can be seen in the figure 1. Here, the star is a non-exposed face.

Definition 1.3**.**

Let $\Upsilon$ be a non-empty collection of convex subsets of $M_{{\mathbb{R}}}$. $\Upsilon$ is called a convex subdivision if the following conditions hold:

1. (1)

Every face of an element of $\Upsilon$ is also in $\Upsilon$. 2. (2)

Every two elements of $\Upsilon$ are either disjoint or they intersect in a common face.

If only (2) is satisfied, then we call $\Upsilon$ a convex decomposition. Let $\Upsilon$ be a convex subdivision or decomposition of $M_{{\mathbb{R}}}$. The support of $\Upsilon$ is the set $|\Upsilon|\coloneqq\bigcup_{C\in\Upsilon}C$. We say $\Upsilon$ is complete if its support is the whole of $M_{{\mathbb{R}}}$. For a given subset $E\subseteq M_{{\mathbb{R}}}$, if $|\Upsilon|=E$, we say $\Upsilon$ is a convex subdivision or decomposition of $E$.

Example 1.4**.**

The set of all faces of a convex set $K$ is a convex subdivision of $K$. The set of all exposed faces of a convex set $K$ is a convex decomposition of $K$.

A concave function is said to be closed if it is upper semicontinuous. This includes the case of continuous, concave functions defined on compact convex sets. The support function of a (not necessarily bounded) convex set $K$ is the function

[TABLE]

given by the assignment

[TABLE]

It is a closed, concave, conical function.

Also, recall that the Legendre–Fenchel dual of a concave function $f\colon N_{{\mathbb{R}}}\to\underline{{\mathbb{R}}}$ is the closed, concave function

[TABLE]

defined by

[TABLE]

whose domain is the so called stability set of $f$, which is denoted by $K_{f}$.

Moreover, the indicator function of a convex set $K\subseteq M_{{\mathbb{R}}}$ is the concave function

[TABLE]

defined by

[TABLE]

The following useful Remark can be found in [BPS14, Section 2.1].

Remark 1.5**.**

Let $K\subseteq M_{{\mathbb{R}}}$ be a convex set and let $\iota_{K}\colon M_{{\mathbb{R}}}\to\underline{{\mathbb{R}}}$ be its indicator function. Then we have that $\phi_{K}=\iota_{K}^{\vee}$ and $\phi_{K}^{\vee}=\iota_{K}$. Hence, the Legendre–Fenchel duality gives a bijective correspondence between indicator functions of compact convex sets in $M_{{\mathbb{R}}}$ and closed, concave, conical functions on $N_{{\mathbb{R}}}$.

Definition 1.6**.**

Let $f$ be a concave function on $N_{{\mathbb{R}}}$. The sup-differential $\partial f(u)$ of $f$ at $u\in N_{{\mathbb{R}}}$ is defined by

[TABLE]

if $f(u)\neq-\infty$, and $\emptyset$ if $f(u)=-\infty$.

This is a generalization to the non-smooth setting of the gradient of a smooth function at a point. Note that in general, the sup-differential may contain more than one point.

Definition 1.7**.**

We say that $f$ is sup-differentiable at a point $u\in N_{{\mathbb{R}}}$ if $\partial f(u)\neq\emptyset$. The effective domain of $f$ is the set of points where $f$ is sup-differentiable. We denote it by ${\operatorname{dom}}(\partial f)$. For a subset $V\subseteq N_{{\mathbb{R}}}$, the set $\partial f(V)$ is defined by

[TABLE]

In particular, the image of $\partial f$ is given by ${\operatorname{Im}}(\partial f)=\partial f(N_{{\mathbb{R}}})$.

The following propositions can be found in [Roc72, Section 30].

Proposition 1.8**.**

The sup-differential $\partial f(u)$ is a closed, convex set for all $u\in{\operatorname{dom}}(\partial f)$. It is bounded if and only if $u\in{\operatorname{relint}}({\operatorname{dom}}(f))$. Moreover, the effective domain of $f$ is close to being convex, in the sense that

[TABLE]

In particular, if ${\operatorname{dom}}(f)=N_{{\mathbb{R}}}$, we have ${\operatorname{dom}}(\partial f)=N_{{\mathbb{R}}}$.

Proposition 1.9**.**

If $f$ is closed, then we have that ${\operatorname{Im}}(\partial f)={\operatorname{dom}}(\partial f^{\vee})$. Hence, the image of the sup-differential is close to being convex, in the sense that

[TABLE]

Definition 1.10**.**

Let $f$ be a closed, concave function on $N_{{\mathbb{R}}}$. We denote by $\Upsilon(f)$ the collection of all sets of the form

[TABLE]

for $m\in{\operatorname{dom}}\left(f^{\vee}\right)\subseteq M_{{\mathbb{R}}}$.

The following is [BPS14, Proposition 2.2.8].

Proposition 1.11**.**

Let $f$ be a closed, concave function on $N_{{\mathbb{R}}}$. Then $\Upsilon(f)$ is a convex decomposition of ${\operatorname{dom}}(\partial f)$. In particular, if ${\operatorname{dom}}(f)=N_{{\mathbb{R}}}$, then $\Upsilon(f)$ is complete.

Definition 1.12**.**

Let $f$ be a closed, concave function on $N_{{\mathbb{R}}}$. The Legendre–Fenchel correspondence of $f$

[TABLE]

is given by the assignment

[TABLE]

Definition 1.13**.**

Let $V$, $V^{*}$ be subsets of $N_{{\mathbb{R}}}$ and of $M_{{\mathbb{R}}}$, respectively. Moreover, let $\Upsilon$, $\Upsilon^{*}$ be convex decompositions of $V$ and $V^{*}$, respectively. We say that $\Upsilon$ and $\Upsilon^{*}$ are dual convex decompositions if there exists a bijective map

[TABLE]

given by the assignment

[TABLE]

and satisfying the following properties:

1. (1)

For every $C,D$ in $\Upsilon$ we have that $C\subseteq D$ if and only if $C^{*}\supseteq D^{*}$. 2. (2)

For every $C$ in $\Upsilon$, the sets $C$ and $C^{*}$ are contained in orthogonal affine spaces of $N_{{\mathbb{R}}}$ and $M_{{\mathbb{R}}}$, respectively.

The following theorem is taken from [BPS14, Theorem 2.2.12].

Theorem 1.14**.**

Let $f$ be a closed, concave function. Then $\mathcal{L}f$ gives a duality between $\Upsilon(f)$ and $\Upsilon(f^{\vee})$ with inverse given by $(\mathcal{L}f)^{-1}=\mathcal{L}f^{\vee}$.

We make the following remark which can be found in [HUL01, Proposition 2.1.5].

Remark 1.15**.**

Consider a convex set $K\subseteq M_{{\mathbb{R}}}$. Let $\phi_{K}$ be the corresponding closed, concave, conical support function and let $C\in\Upsilon(\phi_{K})$. Then, for any $u\in\text{relint}(C)$ we have that $\partial\phi_{K}(u)\in\Upsilon(\phi_{K}^{\vee})$ is an exposed face of $K_{\phi_{K}}=K$. Conversely, every exposed face $F$ of $K$ can be obtained as $\partial\phi_{K}(u)$ for some $u\in N_{{\mathbb{R}}}$. Explicitly, consider $m\in{\operatorname{relint}}(F)$. Then we may take any $u\in{\operatorname{relint}}\left(\partial h^{\vee}_{K}(m)\right)={\operatorname{relint}}\left(\partial\iota_{K}(m)\right)$. In particular, if $K$ is bounded, we get a duality between the set of exposed faces of $K$ and a convex decomposition of $N_{{\mathbb{R}}}$.

Example 1.16**.**

Let notations be as in Remark 1.15 and assume that $K=P$ is a polytope. Then the Legendre–Fenchel duality gives back the classical duality between the faces of a polytope and the cones of its normal fan $\Sigma_{P}$.

If $K$ is not polyhedral, our convex decompositions will not be finite, as can be seen in Figure 2. Here we have

[TABLE]

Note that here the convex decomposition of $N_{{\mathbb{R}}}\simeq{\mathbb{R}}^{2}$ gives us a foliation of the positive quadrant by rays.

2 Canonical decomposition of a difference of convex sets

Throughout this section, $N$ will denote a lattice of rank $n$ and $M=N^{\vee}$ its dual lattice. All convex sets in $M_{{\mathbb{R}}}$ are assumed to be compact, bounded and full-dimensional.

Let $K_{1}\subseteq K_{2}$ be two convex sets in $M_{{\mathbb{R}}}$ with corresponding support functions $\phi_{K_{1}},\phi_{K_{2}}\colon N_{{\mathbb{R}}}\to{\mathbb{R}}$. The aim of this section is to give a canonical decomposition of the difference $K_{2}\setminus K_{1}$.

Definition 2.1**.**

We define two complete convex decompositions $\Sigma_{K_{1}}$ and $\Sigma_{K_{2}}$ of $N_{{\mathbb{R}}}$ by setting

[TABLE]

for $i=1,2$.

Note that the elements in $\Sigma_{K_{i}}$ for $i=1,2$ are cones. This follows from the fact that the convex set $C_{m}$ corresponding to an $m\in{\operatorname{relint}}(K_{i})$ is $\{0\}$. Hence, we will call $\Sigma_{K_{i}}$ a fan, eventhough it may not be finite nor rational.

It follows from Remark 1.15 that the Legendre–Fenchel duality gives an order-reversing, bijective correspondence between cones in $\Sigma_{K_{i}}$ and the set of exposed faces of $K_{i}$ for $i=1,2$. For $F\leq K_{i}$ an exposed face, we will denote by $\sigma_{F}$ the cone in $\Sigma_{K_{i}}$ given by this correspondence.

The following is a key definition for giving the canonical decomposition of the difference $K_{2}\setminus K_{1}$.

Definition 2.2**.**

Let $F_{1}\leq K_{1}$ and $F_{2}\leq K_{2}$ be exposed faces. We say that $F_{1}$ is related to $F_{2}$ (and denote it by $F_{1}\sim F_{2}$) if and only if

[TABLE]

is satisfied.

Definition 2.3**.**

Let $\Sigma\subseteq N_{{\mathbb{R}}}$ be a complete (not necessarily finite nor rational) conical subdivision of $N_{{\mathbb{R}}}$. We say that $\Sigma$ is a difference conical subdivision for $K_{1}$ and $K_{2}$, and denote it by $\Sigma=\Sigma_{K_{2}\setminus K_{1}}$, if the following two conditions are satisfied:

1. (1)

$\Sigma$ is a smooth refinement of both $\Sigma_{K_{1}}$ and $\Sigma_{K_{2}}$. 2. (2)

Let $F_{1}\leq K_{1}$ and $F_{2}\leq K_{2}$ be exposed faces. If $F_{1}\sim F_{2}$, then there exists a $\tau\in\Sigma(1)$ such that $\tau\in\text{relint}(\sigma_{F_{1}})\cap\text{relint}(\sigma_{F_{2}})$.

Remark 2.4**.**

Note that given two $n$-dimensional convex sets $K_{1}\subseteq K_{2}$, we can always find a difference conical subdivision $\Sigma_{K_{1}\setminus K_{2}}$.

The next theorem gives us the canonical decomposition of the difference $K_{2}\setminus K_{1}$. It is the main result of this section.

Theorem 2.5**.**

Let $K_{1}\subseteq K_{2}$ be two $n$-dimensional convex sets in $M_{{\mathbb{R}}}$. Then we have that

[TABLE]

is a convex decomposition of the difference $K_{2}\setminus K_{1}$.

Proof.

Let $F_{1},F_{1}^{\prime}\leq K_{1}$ and $F_{2},F_{2}^{\prime}\leq K_{2}$ be exposed faces such that $F_{1}\sim F_{1}^{\prime}$ and $F_{2}\sim F_{2}^{\prime}$. Now, since $F_{1}\sim F_{2}$ we may fix a $v\in N_{{\mathbb{R}}}$ such that

[TABLE]

Analogously, we may fix $v^{\prime}\in N_{{\mathbb{R}}}$ defining the faces $F_{1}^{\prime}\sim F_{2}^{\prime}$. Note that related faces live in parallel hyperplanes.

Now, let us show that ${\operatorname{convhull}}\left(F_{1},F_{2}\right)\subseteq K_{2}\setminus K_{1}$. Let $m\in{\operatorname{convhull}}(F_{1},F_{2})$. The fact that $m\in K_{2}$ is clear. Now, let $\lambda_{1},\lambda_{2}$ be non-negative real numbers satisfying $\lambda_{1}+\lambda_{2}=1$ and such that

[TABLE]

for $m_{1}\in F_{1}$ and $m_{2}\in F_{2}$. Since $m_{1}\in F_{1}$, $m_{2}\in F_{2}$ and $K_{1}\subseteq K_{2}$, we have

[TABLE]

Hence, we obtain

[TABLE]

which implies that

[TABLE]

in particular ${\operatorname{convhull}}\left(F_{1},F_{2}\right)\subseteq K_{2}\setminus K_{1}$.

Now, let $m^{\prime}\in K_{2}\setminus K_{1}$. Let $F_{2}\leq K_{2}$ be the smallest exposed face such that $m^{\prime}\in F_{2}$. Consider a $\tau\in{\operatorname{relint}}\left(\sigma_{F_{2}}\right)$ and let $F_{1}\leq K_{1}$ be the unique exposed face such that $\tau\in{\operatorname{relint}}\left(\sigma_{F_{1}}\right).$ Then $m^{\prime}\in{\operatorname{convhull}}\left(F_{1},F_{2}\right)$.

Hence, we have shown that

[TABLE]

It remains to show that this is indeed a convex decomposition, i.e. that

[TABLE]

is either empty or a face of both. If the intersection is empty, then we are done. Hence, assume that ${\operatorname{convhull}}\left(F_{1},F_{2}\right)\cap{\operatorname{convhull}}\left(F_{1}^{\prime},F_{2}^{\prime}\right)\neq\emptyset$. By (2.1), the case in which we have that ${\operatorname{convhull}}(F_{1},F_{2})\subseteq{\operatorname{convhull}}(F_{1}^{\prime},F_{2}^{\prime})$ or in which ${\operatorname{convhull}}(F_{1}^{\prime},F_{2}^{\prime})\subseteq{\operatorname{convhull}}(F_{1},F_{2})$ is also clear. Hence, assume that

[TABLE]

and

[TABLE]

Let $H$ be a hyperplane separating $F_{1}$ and $F_{1}^{\prime}$, which exists by the definition of an exposed face. Then, since $F_{1}$ is parallel to $F_{2}$ and $F_{1}^{\prime}$ is parallel to $F_{2}^{\prime}$, we can choose $H$ to be a separating hyperplane of $F_{2}$ and $F_{2}^{\prime}$ as well.

The existence of this separating hyperplane implies that

[TABLE]

which proves that the intersection in (2.2) is a face of both. This concludes the proof of the proposition. ∎

Remark 2.6**.**

As was mentioned in the introduction, in the polyhedral case, the above canonical decomposition a polyhedral subdivision of the complement of two polytopes, one contained in the other. This subdivision appears in the literature (e.g. in [GP88]) although it is constructed using the so called pushing method. We haven’t found in the literature the method we used in Theorem 2.5 nor have we found such a canonical decomposition in the non-polyhedral case.

Let $K_{1}\subseteq K_{2}$ be full-dimensional convex sets in $M_{{\mathbb{R}}}$.

Definition 2.7**.**

Let $F\leq K_{2}$ be an exposed face. To $F$ we associate the correction set

[TABLE]

where for $\tau\in{\operatorname{relint}}\left(\sigma_{F}\right)$, the face $F_{1,\tau}$ is the unique exposed face of $K_{1}$ such that $\tau\in{\operatorname{relint}}\left(\sigma_{F_{1,\tau}}\right)$. The associated correction term $c_{F}$ is defined as $n!$ times the volume of $K_{F}$, i.e.

[TABLE]

Remark 2.8**.**

Note that by Theorem 2.5 we have that

[TABLE]

is a convex decomposition and hence

[TABLE]

Let’s look at a simple $2$-dimensional polyhedral example.

Example 2.9**.**

Consider the simplex $K_{1}$ contained in the square $K_{2}$ as in the Figure 3. Here, the different colors show the correction sets associated to the faces of $K_{2}$. Figure 4 shows the dual picture with the fans $\Sigma_{K_{2}},\,\Sigma_{K_{2}\setminus K_{1}},\,\Sigma_{K_{1}}$.

3 Toric $b$-divisors and surface area measures

Throughout this section, $N$ will denote a lattice of rank $n$ and $M=N^{\vee}$ its dual lattice. All convex sets in $M_{{\mathbb{R}}}$ are assumed to be compact, bounded and full-dimensional.

The goal of this section is to recall the main definitions and facts regarding toric $b$-divisors (see [Bot17]) and to relate the intersection theory of toric $b$-divisors with the so called surface area measure (and a mixed version thereof) associated to a convex set (and to a collection of convex sets) (see [Sch93]).

We fix a complete, smooth fan $\Sigma\subseteq N_{\mathbb{R}}=N\otimes_{{\mathbb{Z}}}{\mathbb{R}}$ and we denote by $X_{\Sigma}$ the corresponding $n$-dimensional, complete, smooth toric variety with dense open torus ${\mathbb{T}}$. We refer to [CLS10] and to [Ful93] for a more detailed introduction to toric geometry. The set $R(\Sigma)$ consists of all smooth sudivisions of $\Sigma$. This is a directed set with partial order given by $\Sigma^{\prime\prime}\geq\Sigma^{\prime}$ in $R(\Sigma)$ if and only if $\Sigma^{\prime\prime}$ is a smooth subdivision of $\Sigma^{\prime}$. The toric Riemann–Zariski space of $X_{\Sigma}$ is defined as the inverse limit

[TABLE]

with maps given by the toric proper birational morphisms $\pi_{\Sigma^{\prime\prime}}\colon X_{\Sigma^{\prime\prime}}\to X_{\Sigma^{\prime}}$ induced whenever $\Sigma^{\prime\prime}\geq\Sigma^{\prime}$. The group of toric Weil $b$-divisors on ${X_{\Sigma}}$ consists of elements in the inverse limit

[TABLE]

where $\mathbb{T}\text{-}{\operatorname{Ca}}(X_{\Sigma^{\prime}})_{\mathbb{Q}}$ denotes the set of toric Cartier ${\mathbb{Q}}$-divisors of $X_{\Sigma^{\prime}}$, with maps given by the push-forward map of toric Cartier ${\mathbb{Q}}$-divisors. We will denote $b$-divisors with bold ${\boldsymbol{D}}$ to distinguish them from classical divisors $D$. We can think of a toric $b$-divisor as a net of toric Cartier ${\mathbb{Q}}$-divisors $\left(D_{\Sigma^{\prime}}\right)_{\Sigma^{\prime}\in R(\Sigma)}$, being compatible under push-forward.

A toric $b$-divisor ${\boldsymbol{D}}=\left(D_{\Sigma^{\prime}}\right)_{\Sigma^{\prime}\in R(\Sigma)}$ is said to be nef, if $D_{\Sigma^{\prime}}\in\mathbb{T}\text{-}{\operatorname{Ca}}(X_{\Sigma^{\prime}})_{\mathbb{Q}}$ is nef for all $\Sigma^{\prime}$ in a cofinal subset of $R(\Sigma)$. It follows from basic toric geometry that there is a bijective correspondence between the set of nef toric $b$-divisors and the set of $\mathbb{Q}$-valued, conical, ${\mathbb{Q}}$-concave functions on $N_{{\mathbb{Q}}}$.

The mixed degree ${\boldsymbol{D}}_{1}\dotsm{\boldsymbol{D}}_{n}$ of a collection of toric $b$-divisors is defined as the limit (in the sense of nets)

[TABLE]

of top intersection numbers of toric divisors, provided this limit exists and is finite. In particular, if ${\boldsymbol{D}}={\boldsymbol{D}}_{1}=\dotsc={\boldsymbol{D}}_{n}$, then the limit (in the sense of nets)

[TABLE]

provided this limit exists and is finite, is called the degree of the toric $b$-divisor ${\boldsymbol{D}}$. A toric $b$-divisor whose degree exists, is said to be integrable.

Let ${\boldsymbol{D}}_{1},\dotsc,{\boldsymbol{D}}_{n}$ be a collection of toric $b$-divisors on a smooth and complete toric variety $X_{\Sigma}$ of dimension $n$ which are nef. Let $\tilde{\phi}_{i}\colon N_{\mathbb{Q}}\to\mathbb{Q}$ be the corresponding $\mathbb{Q}$-concave functions for $i=1,\dotsc,n$. The mixed volume ${\operatorname{MV}}\left(K_{1},\dotsc,K_{n}\right)$ of a collection of convex sets $K_{1},\dotsc,K_{n}$ is defined by

[TABLE]

where the “$+$” refers to Minkowski addition of convex sets. The following theorem relates the mixed degree of nef toric $b$-divisors with the mixed volume of convex bodies. It is a combination of [Bot17, Theorems 4.9 and 4.12].

Theorem 3.1**.**

With notations as above, the functions $\tilde{\phi}_{i}$ extend to continuous, concave functions ${\phi}_{i}\colon N_{\mathbb{R}}\to\mathbb{R}$. Moreover, their mixed degree ${\boldsymbol{D}}_{1}\dotsm{\boldsymbol{D}}_{n}$ exists, and is given by the mixed volume of the stability sets $K_{{\phi}_{i}}$ of the concave functions ${\phi}_{i}$, i.e. we have that

[TABLE]

In particular, a nef toric $b$-divisor ${\boldsymbol{D}}$ is integrable, and its degree is given by

[TABLE]

where ${\phi}$ is the corresponding concave function.

Now, we relate the intersection theory of toric $b$-divisors with the so called surface area measure associated to a convex set (Definition 3.3). We start with some definitions.

Definition 3.2**.**

Consider the vector space ${\mathbb{R}}^{n}$ equipped with the standard euclidean metric. For $0\leq k\leq n$, we let $\mathcal{H}^{k}$ be the $k$-dimensional Hausdorff measure on ${\mathbb{R}}^{n}$. In particular, if $\omega$ is a Borel subset of a $k$-dimensional euclidean space $E^{k}$ or a $k$-dimensional sphere ${\mathbb{S}}^{k}$ in ${\mathbb{R}}^{n}$, then $\mathcal{H}^{k}(\omega)$ coincides with the $k$-dimensional Lebesgue measure of $\omega$ computed in $E^{k}$ or with the $k$-dimensional spherical Lebesgue measure of $\omega$ computed in ${\mathbb{S}}^{k}$, respectively.

Let $K\subseteq{\mathbb{R}}^{n}$ be a compact, bounded, full-dimensional convex set with corresponding support function $\phi_{K}\colon{\mathbb{R}}^{n}\to{\mathbb{R}}$. Moreover, let

[TABLE]

where $\mathcal{P}\left(\partial K\right)$ denotes the power set of the boundary $\partial K$ of $K$, be the map given in the following way. In the case that $\phi_{K}$ is of class $C^{2}$, $g_{K}$ sends $u\in{\mathbb{S}}^{n-1}$ to the gradient $\nabla\phi_{K}(u)$. In general, the inverse $g^{-1}$ is what in the literature is called the Gauss map, which assigns the outer unit normal vector $v_{K}(x)$ to an $x\in\partial_{*}K$, where $\partial_{*}K$ consists of all points in the boundary $\partial K$ of $K$ having a unique outer normal vector. In other words, we have that

[TABLE]

for every $u\in{\mathbb{S}}^{n-1}$. This is also the map given by the Legendre–Fenchel duality (see Section 1).

Definition 3.3**.**

The surface area measure $S_{n-1}(K,\cdot)$ associated to $K$ is the finite Borel measure on the unit sphere ${\mathbb{S}}^{n-1}$ defined by

[TABLE]

for every Borel subset $\omega$ of ${\mathbb{S}}^{n-1}$.

In particular, for a polytope $P$ with unitary normal vectors $u_{1},\dotsc,u_{r}$ at its facets $F_{1},\dotsc,F_{r}$, respectively, the surface area measure of a Borel subset $\omega\subseteq{\mathbb{S}}^{n-1}$ is given by

[TABLE]

where ${\operatorname{vol}}_{k}$ denotes the $k$-dimensional volume operator. In other words, we have that

[TABLE]

where $\delta_{u_{i}}$ denotes the Dirac delta measure supported on $u_{i}\subset{\mathbb{S}}^{n-1}$ for all $i=1,\dotsc,r$.

Example 3.4**.**

Let $P$ be a polytope and let $\phi_{P}$ be its corresponding piecewise linear, concave support function. We get the formula

[TABLE]

for the volume of $P$. This formula can be generalized to any full dimensional bounded, compact, convex set in the following way: for every $n$-dimensional bounded, compact, convex set $K$, the volume of $K$ is given by

[TABLE]

As is described in [Sch93, Section 5], one can generalize the surface area measure associated to a single convex to a collection of $n-1$ (not necessarily distinct) convex sets. This is the so called mixed surface area measure. We denote by $\mathcal{K}_{n}$ the set of compact, bounded, full-dimensional convex sets in ${\mathbb{R}}^{n}$. The following is [Sch93, Theorem 5.1.7].

Theorem/Definition 3.5**.**

There is a nonnegative symmetric function ${\operatorname{V}}\colon\left(\mathcal{K}_{n}\right)^{n}\to{\mathbb{R}}$, called the mixed volume, such that for every natural number $\ell$ and for every non-negative real numbers $\lambda_{1},\dotsc,\lambda_{\ell}$, the equation

[TABLE]

where the sum on the left hand side is the Minkowski sum of convex sets, is satisfied for any collection of convex bodies $K_{1},\dotsc,K_{\ell}\in\mathcal{K}_{n}$.

Furthermore, there is a symmetric map $S$ from $\left(\mathcal{K}_{n}\right)^{n-1}$ into the space of finite Borel measures on ${\mathbb{S}}^{n-1}$, called the mixed surface area measure, such that for every natural number $\ell$ and for every non-negative real numbers $\lambda_{1},\dotsc,\lambda_{\ell}$, the equation

[TABLE]

is satisfied for $K_{1},\dotsc,K_{\ell}\in\mathcal{K}_{n}$ and for every Borel subset $\omega\subseteq\mathbb{S}^{n-1}$. Moreover, for $K_{1},\dotsc,K_{n}\in\mathcal{K}_{n}$, the mixed volume ${\operatorname{V}}\left(K_{1},\dotsc,K_{n}\right)$ can be expressed in terms of the mixed surface area measure in the following way

[TABLE]

We make the following remarks.

Remark 3.6**.**

(1) Setting $K=K_{1}=\dotsb=K_{n}$ we get

[TABLE]

as in Equation (3.3).

(2) The mixed volume “${\operatorname{V}}(\cdot)$” defined above is related to the Mixed Volume “${\operatorname{MV}}(\cdot)$” from Equation 3.1 by the formula

[TABLE]

for $K_{1},\dotsc,K_{n}\in\mathcal{K}_{n}$.

Now, let $K\subseteq M_{{\mathbb{R}}}$ be a convex set. Then the support function $\phi_{K}$ of $K$ corresponds to a nef toric $b$-divisor ${\boldsymbol{D}}_{K}$. The following lemma follows from the definition of the surface area measure and Theorem 3.1.

Lemma 3.7**.**

Let us fix an identification $N_{{\mathbb{R}}}\simeq{\mathbb{R}}^{n}$. Let ${\boldsymbol{D}}_{1},\dotsc,{\boldsymbol{D}}_{n}$ be a collection of nef toric $b$-divisors associated to convex sets $K_{i}\subseteq M_{{\mathbb{R}}}\simeq{\mathbb{R}}^{n}$ with corresponding support functions $\phi_{i}$, for all $i=1,\dotsc,n$. Then the mixed degree ${\boldsymbol{D}}_{1}\dotsm{\boldsymbol{D}}_{m}$ is related to the mixed surface area measure $S_{n-1}(K_{1},\dotsc,K_{n-1},\cdot)$ by the formula

[TABLE]

Moreover, by the symmetry of the mixed surface area measure, we have integral formulae

[TABLE]

for all $i=1,\dotsc,n$.

Remark 3.8**.**

Assuming some smoothness conditions on the support functions of the convex sets, one can compute integrals with respect to (mixed) surface area measure measures explicitly in terms of Lebesgue measures of determinants of Hessians of smooth functions (see [Sch93, Corollary 2.5.3] and the results in [Sch93, Section 5.3]).

4 Volumes and intersection numbers

Throughout this section, $N$ will denote a lattice of rank $n$ and $M=N^{\vee}$ its dual lattice. All convex sets in $M_{{\mathbb{R}}}$ are assumed to be compact, bounded and full-dimensional. We fix two convex sets $K_{1}\subseteq K_{2}$ in $M_{{\mathbb{R}}}$.

The goal of this section is to relate the correction terms of Definition 2.7 with intersection numbers of toric $b$-divisors in the case that $K_{2}$ is a polytope.

Note that two related exposed faces $F_{1}\sim F_{2}$ with $F_{1}\leq K_{1}$ and $F_{2}\leq K_{2}$ are contained in parallel hyperplane sections (defined by the $v$ given in the proof of Theorem 2.5). The following is a key Lemma.

Lemma 4.1**.**

Let $F_{1},\,F_{2}\subseteq{\mathbb{R}}^{d+1}$ be polytopes. Here, $d={\operatorname{max}}\left\{\dim\left(F_{1}\right),\dim\left(F_{2}\right)\right\}$. Assume that $F_{1}\subseteq\{x_{d+1}=0\}$ and that $F_{2}\subseteq\{x_{d+1}=1\}$. Then the volume of the convex hull of $F_{1},F_{2}$ is given by

[TABLE]

Proof.

We start with the following three claims:

Claim 1: Let $\lambda$ be any real number between [math] and $1$. Then the slice of the convex hull of $F_{1}$ and $F_{2}$ at $x_{d+1}=\lambda$ is given by

[TABLE]

where the sum in the right hand side is the Minkowski sum of convex sets.

Claim 2: Let $\lambda$ be any real number between [math] and $1$. Then it follows that the volume of the slice $\lambda F_{1}+(1-\lambda)F_{2}\subseteq{\operatorname{convhull}}\left(F_{1},F_{2}\right)$ is given by

[TABLE]

Claim 3: Let $\lambda$ be as before and let $\ell,\,k$ be two non-negative integers with $k\leq\ell$. We define the number ${\operatorname{I}}(\ell,k)$ by

[TABLE]

Then the formula

[TABLE]

holds true.

Now, Claim 1 is clear and Claim 2 is a standard result in convex geometry. We proceed to give a proof of Claim 3: integrating by parts, we get

[TABLE]

Moreover the values for $k=\ell$ and for $k=0$ are given by

[TABLE]

Hence, we get

[TABLE]

as we wanted to show.

Finally, note that Claim 1, Claim 2 and Claim 3 imply that

[TABLE]

concluding the proof of the lemma. ∎

Consider the convex sets $K_{1}\subseteq K_{2}$ with corresponding support functions $\phi_{1},\phi_{2}$. Moreover, let $\Sigma=\Sigma_{K_{2}\setminus K_{1}}\subseteq N_{{\mathbb{R}}}$ be a difference conical subdivision.

We have the following theorem.

Theorem 4.2**.**

Let notations be as above and assume that $K_{2}$ is a polytope. Then the functions $\phi_{1}$ and $\phi_{2}$ correspond respectively to nef toric $b$-divisors ${\boldsymbol{D}}_{1}$ and ${\boldsymbol{D}}_{2}$ on the normal fan of $K_{2}$, denoted by $\Sigma_{K_{2}}$. We can express the difference ${\boldsymbol{D}}_{2}^{n}-{\boldsymbol{D}}_{1}^{n}$ of degrees of the nef toric $b$-divisors ${\boldsymbol{D}}_{1}$ and ${\boldsymbol{D}}_{2}$ as a finite sum of correction terms

[TABLE]

where the correction terms $c_{F}$ are related to top intersection numbers of toric $b$-divisors by

[TABLE]

where $S(\cdot)$ is the mixed surface area measure defined in the previous section. In particular, if $K_{1}$ is also polyhedral, then $\Sigma$ is a real rational, polyhedral fan and we get

[TABLE]

where $D_{r}$ is the divisor corresponding to the ray $r\in\Sigma(1)$ and all the intersection products are done in $\Sigma$.

Proof.

The first and last statement of the theorem follow from Theorem 2.5 and Definition 2.7. For the statement regarding the expression of the correction terms in terms of intersection numbers, we have

[TABLE]

concluding the proof of the theorem.

∎

Example 4.3**.**

Consider the fan of ${\mathbb{P}}^{2}$ and the nef toric $b$-divisors $\phi_{1}$ and $\phi_{2}$ given by the concave functions $\phi_{1},\phi_{2}\colon{\mathbb{R}}^{2}\to{\mathbb{R}}$ defined by

[TABLE]

and

[TABLE]

and consider the corresponding convex sets $K_{1}\subseteq K_{2}$. Note that $K_{2}$ is the $2$-dimensional symplex and $K_{1}$ is the convex set from Example 1.16. The only face of the simplex $K_{2}$ whose associated correction term is non-zero is the vertex $F_{0}$ in Figure 5.

On the one hand, we can calculate the difference $\phi_{2}^{2}-\phi_{1}^{2}$ as the difference of volumes of convex sets

[TABLE]

On the other hand, Theorem 4.2 tells us that we can compute the correction term $c_{F_{0}}$ by

[TABLE]

where the last equality follows from a computation using Remark 3.8.