Maximum and minimum of support functions

TL;DR

This paper investigates how maximum and minimum support functions relate to Wulff shapes, showing that the shape for the maximum is the convex hull of the union, and for the minimum is the intersection, with dual relationships explored.

Contribution

It establishes geometric relationships between support functions and Wulff shapes for maximum and minimum functions, extending understanding of convex integrands.

Findings

Wulff shape for max support function is convex hull of union.

Wulff shape for min support function is intersection of shapes.

Dual relationships between Wulff shapes are characterized.

Abstract

For given continuous functions $\gamma_{{}_{i}}: S^{n}\to \mathbb{R}_{+}$ (where $i=1, 2$), the functions $\gamma_{{}_{max}}$ and $\gamma_{{}_{min}}$ can be defined as natural way. In this paper, we show that the Wulff shape associated to $\gamma_{{}_{max}}$ is the convex hull of the union of Wulff shapes associated to $\gamma_{{}_1}$ and $\gamma_{{}_2}$ , if $\gamma_{{}_1}$ and $\gamma_{{}_2}$ are convex integrands. And, the Wulff shape associated to $\gamma_{{}_{min}}$ is the intersection of Wulff shapes associated to $\gamma_{{}_1}$ and $\gamma_{{}_2}$. Moreover, relationships between their dual Wulff shapes are given.

Equations

$${\mathcal{W}}_{\gamma }=\bigcap _{\theta \in S^n}{\Gamma }_{\gamma ,\theta }.$$

$${\Gamma }_{\gamma ,\theta }=\{x\in {\mathbb{R}}^{n+1}|x\cdot \theta \le \gamma (\theta )\},$$

$$\text{mbox}graph(\gamma )=\{(\theta ,\gamma (\theta ))\in {\mathbb{R}}^{n+1}-\{0\}|\theta \in S^n\},$$

$$\text{mbox}inv(\theta ,r)=\left(-\theta ,\frac{1}{r}\right).$$

$${\gamma }_{{}_{max}}:S^n\to {\mathbb{R}}_{+},{\gamma }_{{}_{max}}(\theta )=\max \{{\gamma }_{{}_1}(\theta ),{\gamma }_{{}_2}(\theta )\}.$$

$${\gamma }_{{}_{min}}:S^n\to {\mathbb{R}}_{+},{\gamma }_{{}_{min}}(\theta )=\min \{{\gamma }_{{}_1}(\theta ),{\gamma }_{{}_2}(\theta )\}.$$

$${\mathcal{W}}_{{\gamma }_{{}_{max}}}=\mathrm{convex}\mathrm{hull}\mathrm{of}\mathrm{the}({\mathcal{W}}_{{\gamma }_{{}_1}}\cup {\mathcal{W}}_{{\gamma }_{{}_2}}).$$

$${\mathcal{W}}_{{\gamma }_{{}_{min}}}={\mathcal{W}}_{{\gamma }_{{}_1}}\cap {\mathcal{W}}_{{\gamma }_{{}_2}}.$$

$$H(\tilde{P})=\{\tilde{Q}\in S^{n+1}\mid \tilde{P}\cdot \tilde{Q}\ge 0\},$$

$${\tilde{W}}^{\circ }=\bigcap _{\tilde{P}\in \tilde{W}}H(\tilde{P}).$$

$$\tilde{P}\tilde{Q}=\left\{\frac{(1-t)\tilde{P}+t\tilde{Q}}{||(1-t)\tilde{P}+t\tilde{Q}||}\in S^{n+1}|0\le t\le 1\right\}.$$

$$\mbox{\rm s-conv}(\widetilde{W})=\left\{\left.\frac{\sum_{i=1}^{k}t_{i}\widetilde{P}_{i}}{||\sum_{i=1}^{k}t_{i}\widetilde{P}_{i}||}\;\right|\;\widetilde{P}_{i}\in{\color[rgb]{0,0,0}\widetilde{W}},\;\sum_{i=1}^{k}t_{i}=1,\;t_{i}\geq 0,k\in\mathbb{N}\right\}.$$

$${\left\{\frac{{\sum }_{i=1}^k{t}_i{\tilde{P}}_i}{||{\sum }_{i=1}^k{t}_i{\tilde{P}}_i||}|{\tilde{P}}_i\in X,\sum _{i=1}^k{t}_i=1,t_i\ge 0\right\}}^{\circ }=H({\tilde{P}}_1)\cap \cdots \cap H({\tilde{P}}_k).$$

$${\alpha }_{{}_N}\left(P_1,\dots ,P_{n+1},P_{n+2}\right)=\left(\frac{P_1}{P_{n+2}},\dots ,\frac{P_{n+1}}{P_{n+2}},1\right).$$

$${\Psi }_N(\tilde{P})=\frac{1}{\sqrt{1-(N\cdot P{)}^2}}(N-(N\cdot \tilde{P})\tilde{P}).$$

$${\mathcal{H}}^{\circ }(S^{n+1})=\{\tilde{W}\in \mathcal{H}(S^{n+1})\mid {\tilde{W}}^{\circ }\ne \varnothing \}.$$

$${\mathcal{W}}_{\gamma }=I{d}^{-1}\circ {\alpha }_{{}_N}\left({\left({\Psi }_N\circ {\alpha }_{{}_N}^{-1}\circ Id\left(\text{mbox}\mathrm{graph}(\gamma )\right)\right)}^{\circ }\right).$$

$$\left(\Psi_{N}\circ\alpha_{{}_{N}}^{-1}\circ Id\left(\mbox{\rm graph}(\gamma_{{}_{i}})\right)\right)^{\circ}=\left(\mbox{s-conv}\bigl{(}\Psi_{N}\circ\alpha_{{}_{N}}^{-1}\circ Id\left(\mbox{\rm graph}(\gamma_{{}_{i}})\right)\bigr{)}\right)^{\circ},$$

$$I{d}^{-1}\circ {\alpha }_{{}_N}\left({\left({\alpha }_{{}_N}^{-1}\circ Id\left({\mathcal{W}}_{\gamma }\right)\right)}^{\circ }\right).$$

$${\mathcal{W}}_{{\gamma }_{{}_{max}}}$$

$$=I{d}^{-1}\circ {\alpha }_{{}_N}\left((\partial (\mathcal{D}{\tilde{\mathcal{W}}}_{{\gamma }_{{}_1}}\cap \mathcal{D}{\tilde{\mathcal{W}}}_{{\gamma }_{{}_2}}){)}^{\circ }\right)$$

$$=I{d}^{-1}\circ {\alpha }_{{}_N}\left((\mathcal{D}{\tilde{\mathcal{W}}}_{{\gamma }_{{}_1}}\cap \mathcal{D}{\tilde{\mathcal{W}}}_{{\gamma }_{{}_2}}{)}^{\circ }\right)$$

$${\tilde{\mathcal{W}}}_{{\gamma }_{{}_i}}={\alpha }_{{}_N}^{-1}\circ Id({\mathcal{W}}_{{\gamma }_{{}_i}})$$

$$(\mathcal{D}{\tilde{\mathcal{W}}}_{{\gamma }_{{}_1}}\cap \mathcal{D}{\tilde{\mathcal{W}}}_{{\gamma }_{{}_2}}{)}^{\circ }=\text{mbox}s-conv({\tilde{\mathcal{W}}}_{{\gamma }_{{}_1}}\cup {\tilde{\mathcal{W}}}_{{\gamma }_{{}_2}}).$$

$${\tilde{\mathcal{W}}}_{{\gamma }_{{}_1}}\subset H(\tilde{P})\mathrm{and}{\tilde{\mathcal{W}}}_{{\gamma }_{{}_2}}\subset H(\tilde{P}).$$

$$({\tilde{\mathcal{W}}}_{{\gamma }_{{}_1}}\cup {\tilde{\mathcal{W}}}_{{\gamma }_{{}_2}}{)}^{\circ }\subset {\tilde{\mathcal{W}}}_{{\gamma }_{{}_1}}^{\circ }\cap {\tilde{\mathcal{W}}}_{{\gamma }_{{}_2}}^{\circ }=\mathcal{D}{\tilde{\mathcal{W}}}_{{\gamma }_{{}_1}}\cap \mathcal{D}{\tilde{\mathcal{W}}}_{{\gamma }_{{}_2}}.$$

$$\bigl{(}\mbox{s-conv}(\widetilde{\mathcal{W}}_{\gamma_{{}_{1}}}\cup\widetilde{\mathcal{W}}_{\gamma_{{}_{2}}})\bigr{)}^{\circ}=(\widetilde{\mathcal{W}}_{\gamma_{{}_{1}}}\cup\widetilde{\mathcal{W}}_{\gamma_{{}_{2}}})^{\circ}\subset\mathcal{D}\widetilde{\mathcal{W}}_{\gamma_{{}_{1}}}\cap\mathcal{D}\widetilde{\mathcal{W}}_{\gamma_{{}_{2}}}.$$

$$\bigl{(}\mathcal{D}\widetilde{\mathcal{W}}_{\gamma_{{}_{1}}}\cap\mathcal{D}\widetilde{\mathcal{W}}_{\gamma_{{}_{2}}}\bigr{)}^{\circ}\subset\bigl{(}\mbox{s-conv}(\widetilde{\mathcal{W}}_{\gamma_{{}_{1}}}\cup\widetilde{\mathcal{W}}_{\gamma_{{}_{2}}})\bigr{)}^{\circ\circ}=\mbox{s-conv}(\widetilde{\mathcal{W}}_{\gamma_{{}_{1}}}\cup\widetilde{\mathcal{W}}_{\gamma_{{}_{2}}}).$$

$$(\mathcal{D}{\tilde{\mathcal{W}}}_{{\gamma }_{{}_1}}\cap \mathcal{D}{\tilde{\mathcal{W}}}_{{\gamma }_{{}_2}})\subset {\tilde{W}}_{{\gamma }_{{}_1}}^{\circ }.$$

$${\tilde{W}}_{{\gamma }_{{}_1}}={\tilde{W}}_{{\gamma }_{{}_1}}^{\circ \circ }\subset (\mathcal{D}{\tilde{\mathcal{W}}}_{{\gamma }_{{}_1}}\cap \mathcal{D}{\tilde{\mathcal{W}}}_{{\gamma }_{{}_2}}{)}^{\circ }.$$

$${\tilde{W}}_{{\gamma }_{{}_2}}\subset (\mathcal{D}{\tilde{\mathcal{W}}}_{{\gamma }_{{}_1}}\cap \mathcal{D}{\tilde{\mathcal{W}}}_{{\gamma }_{{}_2}}{)}^{\circ }.$$

Full text

1. Introduction

Let $n$ be a positive integer. Given a continuous function $\gamma:S^{n}\to\mathbb{R}_{+}$, where $S^{n}$ is the unit sphere in $\mathbb{R}^{n+1}$ and $\mathbb{R}_{+}$ is the set consisting of positive real numbers, the Wulff shape associated associated with the support function $\gamma$, denoted by $\mathcal{W}_{\gamma}$, is the following intersection (see Figure 1),

[TABLE]

Here, $\Gamma_{\gamma,\theta}$ is the following half-space:

[TABLE]

where the dot in the center stands for the dot product of two vectors $x,\theta\in\mathbb{R}^{n+1}$.

This construction is well-known as Wulff’s construction of an equilibrium crystal introduced by G. Wulff in [18] (for details on Wulff shapes, see for instance [1, 12, 15, 16, 17]). By definition, a Wulff shape is convex, compact and it contains the origin of $\mathbb{R}^{n+1}$ as an interior point. Conversely, it has been known that any convex body $W$ in $\mathbb{R}^{n+1}$ containing the origin as an interior point is a Wulff shape associated with an appropriate support function, namely, there exists a continuous function $\gamma:S^{n}\to\mathbb{R}_{+}$ such that $W=\mathcal{W}_{\gamma}$ ([16]).

For a continuous function $\gamma:S^{n}\to\mathbb{R}_{+}$, set

[TABLE]

where $(\theta,\gamma(\theta))$ is the polar plot expression for a point of $\mathbb{R}^{n+1}-\{0\}$. The mapping $\mbox{inv}:\mathbb{R}^{n+1}-\{0\}\to\mathbb{R}^{n+1}-\{0\}$, defined as follows, is called the inversion with respect to the origin of $\mathbb{R}^{n+1}$.

[TABLE]

Let $\Gamma_{\gamma}$ be the boundary of the convex hull of $\mbox{inv}(\mbox{graph}(\gamma))$. If the equality $\Gamma_{\gamma}=\mbox{inv}(\mbox{graph}(\gamma))$ is satisfied, then $\gamma$ is called a convex integrand (see Figure 6).

The notion of convex integrand was first introduced by J. Taylor in [16] and it plays a key role for studying Wulff shapes (for details on convex integrands, see for instance [3, 12]). For given support functions $\gamma_{{}_{1}},\gamma_{{}_{2}}$, define $\gamma_{{}_{max}}$ and $\gamma_{{}_{min}}$ as follows.

[TABLE]

[TABLE]

Then the question naturally arises “What are the relationships between $\mathcal{W}_{\gamma_{{}_{1}}},\mathcal{W}_{\gamma_{{}_{2}}}$ and $\mathcal{W}_{\gamma_{{}_{max}}}$ (or $\mathcal{W}_{\gamma_{{}_{min}}}$)? How they are related ?”. The main results of this paper are as follows (see Figure 3).

Theorem 1**.**

Let $\gamma_{{}_{1}}$ and $\gamma_{{}_{2}}$ be convex integrands. Then the following holds:

[TABLE]

Theorem 2**.**

Let $\gamma_{{}_{1}}$ and $\gamma_{{}_{2}}$ be support(continuuos) functions. Then the following holds:

[TABLE]

This paper is organized as follows. In section 2, the preliminaries are given, and the proof of Theorem 1, Theorem 2 and related topics are given in section 3, 4 and section 5 respectively.

2. Preliminaries

2.1. Spherical convex body

For any point $\widetilde{P}\in S^{n+1}$, let $H(\widetilde{P})$ be the hemisphere centered at $\widetilde{P}$,

[TABLE]

where the dot in the center stands for the scalar product of two vectors $\widetilde{P},\widetilde{Q}\in\mathbb{R}^{n+2}$. For any non-empty subset $\widetilde{W}\subset S^{n+1}$, the spherical polar set of $\widetilde{W}$, denoted by $\widetilde{W}^{\circ}$, is defined as follows:

[TABLE]

Definition 1** ([14]).**

Let $\widetilde{W}$ be a subset of $S^{n+1}$. Suppose that there exists a point $\widetilde{P}\in S^{n+1}$ such that $\widetilde{W}\cap H(\widetilde{P})=\emptyset$. Then, $\widetilde{W}$ is said to be hemispherical.

Let $\widetilde{P},\widetilde{Q}$ be two points of $S^{n+1}$ such that $(1-t)\widetilde{P}+t\widetilde{Q}$ is not the zero vector for any $t\in[0,1]$. Then, the following arc is denoted by $\widetilde{P}\widetilde{Q}$:

[TABLE]

Definition 2** ([14]).**

Let $\widetilde{W}\subset S^{n+1}$ be a hemispherical subset.

1. (1)

Suppose that $\widetilde{P}\widetilde{Q}\subset\widetilde{W}$ for any $\widetilde{P},\widetilde{Q}\in\widetilde{W}$. Then, $\widetilde{W}$ is said to be spherical convex. 2. (2)

Suppose that $\widetilde{W}$ is closed, spherical convex and has an interior point. Then, $\widetilde{W}$ is said to be a spherical convex body.

Definition 3** ([14]).**

Let $\widetilde{W}$ be a hemispherical subset of $S^{n+1}$. Then, the following set, denoted by \mbox{\rm s-conv}({\color[rgb]{0,0,0}\widetilde{W}}), is called the spherical convex hull of {\color[rgb]{0,0,0}\widetilde{W}}.

[TABLE]

Lemma 2.1** ([14]).**

For any hemispherical subset $\widetilde{W}$ of $S^{n+1}$, the spherical convex hull of $\widetilde{W}$ is the smallest spherical convex set containing $\widetilde{W}$.

Lemma 2.2** ([14]).**

Let $\widetilde{W}_{1},\widetilde{W}_{2}\subset S^{n+1}$. If $\widetilde{W}_{1}$ is a subset of $\widetilde{W}_{2}$, then $\widetilde{W}_{2}^{\circ}$ is a subset of $\widetilde{W}_{1}^{\circ}$.

Lemma 2.3** ([14]).**

For any subset $\widetilde{W}$ of $S^{n+1}$, the inclusion $\widetilde{W}\subset\widetilde{W}^{\circ\circ}$ holds.

The following proposition has been known.

Proposition 1** ([14]).**

For any non-empty closed hemispherical subset $\widetilde{W}\subset S^{n+1}$, the equality $\mbox{\rm s-conv}(\widetilde{W})=(\mbox{\rm s-conv}(\widetilde{W}))^{\circ\circ}$ holds.

Lemma 2.4** (Maehara’s lemma ([11, 14])).**

For any hemispherical finite subset $\widetilde{W}=\{\widetilde{P}_{1},\ldots,\widetilde{P}_{k}\}\subset S^{n+1}$, the following holds:

[TABLE]

Maeara’s lemma was first given in [11], which is a useful tool to study spherical convex bodies. For details, see for instance [14].

2.2. An equivalent definition

In [14], an equivalent definition of Wulff shape has been given, which is defined as the composition of the following mappings.

1. Mapping $Id:\mathbb{R}^{n+1}\to\mathbb{R}^{n+1}\times\{1\}$

The mapping $Id$ defined by $Id(x)=(x,1)$.

2. Central projection $\alpha_{{}_{N}}:S_{N,+}^{n+1}\to\mathbb{R}^{n+1}\times\{1\}$

Denote the point $(0,\ldots,0,1)\in\mathbb{R}^{n+2}$ by $N$. The set $S^{n+1}-H(-N)$ is denoted by $S_{N,+}^{n+1}$. Let $\alpha_{{}_{N}}:S_{N,+}^{n+1}\to\mathbb{R}^{n+1}\times\{1\}$ be the central projection relative to $N$, namely, $\alpha_{N}$ is defined as follows for any $(P_{1},\ldots,P_{n+1},P_{n+2})\in S_{N,+}^{n+1}$:

[TABLE]

Through out remainder of this paper, let $\widetilde{X}=\alpha_{{}_{N}}^{-1}\circ Id(X)$ for any non-empty subset $X$ of $\mathbb{R}^{n+1}$. For any Wulfff shape $\mathcal{W}\subset\mathbb{R}^{n+1}$, the spherical convex body $\widetilde{\mathcal{W}}=\alpha_{{}_{N}}^{-1}\circ Id(\mathcal{W})$ is called spherical Wulff shape of $\mathcal{W}$. Then the following two are equivalent (see [14] for details ).

1. (1)

$\widetilde{\mathcal{W}}$ is a spherical Wulff shape. 2. (2)

$\widetilde{\mathcal{W}}$ is a spherical convex body such that $\widetilde{\mathcal{W}}\cap H(-N)=\emptyset$ and $N$ is an interior point of $\widetilde{\mathcal{W}}$.

3. Spherical blow-up \Psi_{N}:S^{n+1}-\{\pm N\}\to{\color[rgb]{0,0,0}S_{N,+}^{n+1}}

Next, we consider the mapping \Psi_{N}:S^{n+1}-\{\pm N\}\to{\color[rgb]{0,0,0}S_{N,+}^{n+1}} defined by

[TABLE]

The mapping $\Psi_{N}$, which was first introduced in [13], has the following intriguing properties:

1. (1)

For any $\widetilde{P}\in S^{n+1}-\{\pm N\}$, the equality $\widetilde{P}\cdot\Psi_{N}(\widetilde{P})=0$ holds, 2. (2)

for any $\widetilde{P}\in S^{n+1}-\{\pm N\}$, the property $\Psi_{N}(\widetilde{P})\in\mathbb{R}N+\mathbb{R}\widetilde{P}$ holds, 3. (3)

for any $\widetilde{P}\in S^{n+1}-\{\pm N\}$, the property $N\cdot\Psi_{N}(\widetilde{P})>0$ holds, 4. (4)

the restriction $\Psi_{N}|_{S^{n+1}_{N,+}-\{N\}}:S^{n+1}_{N,+}-\{N\}\to S^{n+1}_{N,+}-\{N\}$ is a $C^{\infty}$ diffeomorphism.

4. Spherical polar transform $\bigcirc:\mathcal{H}^{\circ}(S^{n+1})\to\mathcal{H}^{\circ}(S^{n+1})$

Let $\mathcal{H}(S^{n+1})$ be the set consisting of non-empty compact set of $S^{n+1}$. It is clear that the spherical polar set of $S^{n+1}$ is the empty set. Let $\mathcal{H}^{\circ}(S^{n+1})$ be the subspace of $\mathcal{H}(S^{n+1})$ defined as follows.

[TABLE]

The spherical polar transform $\bigcirc:\mathcal{H}^{\circ}(S^{n+1})\to\mathcal{H}^{\circ}(S^{n+1})$ is defined by $\bigcirc(\widetilde{W})=\widetilde{W}^{\circ}.$ Since $\widetilde{W}\subset\widetilde{W}^{\circ\circ}$ for any $\widetilde{W}\in\mathcal{H}^{\circ}(S^{n+1})$ by Lemma 2.3, it follows that $\widetilde{W}^{\circ}\in\mathcal{H}^{\circ}(S^{n+1})$ for any $\widetilde{W}\in\mathcal{H}^{\circ}(S^{n+1})$. Thus, the spherical polar transform is well-defined. It is known that the spherical polar transform is Lipchitz with respect to the Pompeiu-Hausdorff distance([2]). Moreover, the restriction of the spherical polar transform to the set consisting of spherical Wulff shape relative to $\widetilde{P}$ (see Definition 5) is an isometry with respect to the Pompeiu-Hausdorff distance ([2]).

Proposition 2** ([14]).**

Let $\gamma:S^{n}\to\mathbb{R}_{+}$ be a continuous function. Then, $\mathcal{W}_{\gamma}$ is characterized as follows:

[TABLE]

Proposition 2 implies Wulff shapes $\mathcal{W}_{\gamma_{{}_{1}}}$ and $\mathcal{W}_{\gamma_{{}_{2}}}$ are same convex bodies if and only if $\left(\Psi_{N}\circ\alpha_{{}_{N}}^{-1}\circ Id\left(\mbox{\rm graph}(\gamma_{{}_{1}})\right)\right)^{\circ}$ and $\left(\Psi_{N}\circ\alpha_{{}_{N}}^{-1}\circ Id\left(\mbox{\rm graph}(\gamma_{{}_{2}})\right)\right)^{\circ}$ are same spherical convex bodies. By Maehara’s lemma (Lemma 2.4), we know that

[TABLE]

where $i=1,2.$ Therefore, the following holds:

Proposition 3** ([14]).**

Let $\gamma_{{}_{i}}:S^{n}\to\mathbb{R}_{+}$ be continuous function, where $i=1,2$. Then the following two statements are equivalent:

1. (1)

$\mathcal{W}_{\gamma_{{}_{1}}}=\mathcal{W}_{\gamma_{{}_{2}}}$. 2. (2)

\mbox{\rm s-conv}\bigl{(}\Psi_{N}\circ\alpha_{{}_{N}}^{-1}\circ Id\left(\mbox{\rm graph}(\gamma_{{}_{1}})\right)\bigr{)}=\mbox{\rm s-conv}\bigl{(}\Psi_{N}\circ\alpha_{{}_{N}}^{-1}\circ Id\left(\mbox{\rm graph}(\gamma_{{}_{2}})\right)\bigr{)}.

Proposition 4** ([14]).**

For any Wulff shape $\mathcal{W}_{\gamma}$, the following set, too, is a Wulff shape:

[TABLE]

Definition 4** ([14]).**

For any Wulff shape $\mathcal{W}_{\gamma}$, the Wulff shape given in Proposition 4 is called the dual Wulff shape of $\mathcal{W}_{\gamma}$ and is denoted by $\mathcal{D}\mathcal{W}_{\gamma}$. **

By Proposition 4, the following definition is reasonable.

Definition 5** ([14]).**

Let $\widetilde{P}$ be a point of $S^{n+1}$.

1. (1)

A spherical convex body $\widetilde{W}$ such that $\widetilde{W}\cap H(-\widetilde{P})=\emptyset$ and $\widetilde{P}\in\mbox{\rm int}(\widetilde{W})$ are satisfied is called a spherical Wulff shape relative to $\widetilde{P}$. 2. (2)

Let $\widetilde{\mathcal{W}}$ be a spherical Wulff shape relative to $\widetilde{P}$. Then, the set $\widetilde{\mathcal{W}}^{\circ}$ is called the spherical dual Wulff shape of the spherical Wulff shape $\widetilde{\mathcal{W}}$ relative to $\widetilde{P}$ and is denoted by $\mathcal{D}\widetilde{\mathcal{W}}$.

Proposition 5** ([14]).**

Let $\bar{\gamma}:S^{n}\to\mathbb{R}_{+}$ be a convex integrand. Let $\mathcal{D}\mathcal{W}_{\gamma}$ be the dual Wulff shape of $\mathcal{W}_{\gamma}.$ Then the boundary of the $\mathcal{D}\mathcal{W}_{\gamma}$ is exactly $\Gamma_{\gamma}=\mbox{inv}(\mbox{graph}(\gamma)).$

Proposition 2 gives a new powerful spherical method to study Wulff shapes. For example, the self-dual Wulff shape $\mathcal{W}_{\gamma}=\mathcal{DW}_{\gamma}$ can be characterized by the induced spherical convex body of constant width ${\pi}/{2}$ ([4]). More details on width of spherical convex bodies and their duals, see for instance [6]–[10]. For related topics on the spherical method see for instance [5].

3. Proof of Theorem 1

By Proposition 2, $\mathcal{W}_{\gamma_{{}_{max}}}$ can be rewritten as

[TABLE]

where $\partial(\mathcal{D}\widetilde{\mathcal{W}}_{\gamma_{{}_{1}}}\cap\mathcal{D}\widetilde{\mathcal{W}}_{\gamma_{{}_{2}}})$ is the boundary of $(\mathcal{D}\widetilde{\mathcal{W}}_{\gamma_{{}_{1}}}\cap\mathcal{D}\widetilde{\mathcal{W}}_{\gamma_{{}_{2}}})$, and

[TABLE]

is the spherical Wulff shape of $\mathcal{W}_{\gamma_{{}_{i}}},i=1,2$. Here, the second equality follows from Proposition 5 and the third equality follows from Maehara’s lemma (Lemma 2.4). Thus it is sufficient to prove the following:

[TABLE]

First, we show that $(\mathcal{D}\widetilde{\mathcal{W}}_{\gamma_{{}_{1}}}\cap\mathcal{D}\widetilde{\mathcal{W}}_{\gamma_{{}_{2}}})^{\circ}\subset\mbox{s-conv}(\widetilde{\mathcal{W}}_{\gamma_{{}_{1}}}\cup\widetilde{\mathcal{W}}_{\gamma_{{}_{2}}})$. Let $\widetilde{P}$ be a point of $(\widetilde{\mathcal{W}}_{\gamma_{{}_{1}}}\cup\widetilde{\mathcal{W}}_{\gamma_{{}_{2}}})^{\circ}$. Then it follows that

[TABLE]

Thus $\widetilde{P}$ is a point of $\widetilde{\mathcal{W}}_{\gamma_{{}_{1}}}^{\circ}\cap\widetilde{\mathcal{W}}_{\gamma_{{}_{2}}}^{\circ}=\mathcal{D}\widetilde{\mathcal{W}}_{\gamma_{{}_{1}}}\cap\mathcal{D}\widetilde{\mathcal{W}}_{\gamma_{{}_{2}}}$. This implies that

[TABLE]

Then by Maehara’s lemma, we have

[TABLE]

Since \bigl{(}\mbox{s-conv}(\widetilde{\mathcal{W}}_{\gamma_{{}_{1}}}\cup\widetilde{\mathcal{W}}_{\gamma_{{}_{2}}})\bigr{)}^{\circ\circ}=\mbox{s-conv}(\widetilde{\mathcal{W}}_{\gamma_{{}_{1}}}\cup\widetilde{\mathcal{W}}_{\gamma_{{}_{2}}}) (Proposition 1), by Lemma 2.2, it follows that

[TABLE]

Next, we show that $\mbox{s-conv}(\widetilde{\mathcal{W}}_{\gamma_{{}_{1}}}\cup\widetilde{\mathcal{W}}_{\gamma_{{}_{2}}})\subset(\mathcal{D}\widetilde{\mathcal{W}}_{\gamma_{{}_{1}}}\cap\mathcal{D}\widetilde{\mathcal{W}}_{\gamma_{{}_{2}}})^{\circ}$. Since $\mathcal{D}\widetilde{\mathcal{W}}_{\gamma_{{}_{1}}}\cap\mathcal{D}\widetilde{\mathcal{W}}_{\gamma_{{}_{2}}}$ is a subset of $\mathcal{D}\widetilde{\mathcal{W}}_{\gamma_{{}_{1}}}$, by Lemma 2.2, we have

[TABLE]

Then by Proposition 1, it follows that

[TABLE]

In the same way, the following inclusion is holds:

[TABLE]

Since $\mbox{s-conv}(\widetilde{\mathcal{W}}_{\gamma_{{}_{1}}}\cup\widetilde{\mathcal{W}}_{\gamma_{{}_{2}}})$ is the smallest convex body containing $\widetilde{\mathcal{W}}_{\gamma_{{}_{1}}}\cup\widetilde{\mathcal{W}}_{\gamma_{{}_{2}}}$ (Lemma 2.1) and $(\mathcal{D}\widetilde{\mathcal{W}}_{\gamma_{{}_{1}}}\cap\mathcal{D}\widetilde{\mathcal{W}}_{\gamma_{{}_{2}}})^{\circ}$ is a convex body, it follows that

[TABLE]

This proves the Theorem. $\square$

Remark that the condition $``$convex integrand” of Theorem 1 is necessary. Theorem 1 does not hold in general (see Figure 4 and 5).

Usually, for given convex integrands $\gamma_{{}_{1}}$ and $\gamma_{{}_{2}}$, the inversion of $\mbox{graph}(\gamma_{{}_{min}})$ with respect to the origin of $\mathbb{R}^{n+1}$ does not necessarily equal to $\Gamma_{\gamma_{{}_{min}}}$. Thus the function $\gamma_{{}_{min}}$ is not convex integrand in general. On the other hand, the inversion of $\mbox{graph}(\gamma_{{}_{max}})$ with respect to the origin of $\mathbb{R}^{n+1}$ is exactly $\Gamma_{\gamma_{{}_{max}}}$, which implies that $\gamma_{{}_{max}}$ is a convex integrand. Thus we can generalize Theorem 1 as follows.

Corollary 1**.**

Let $\gamma_{{}_{1}},\dots,\gamma_{{}_{m}}$ be convex integrands. Let $\widetilde{\gamma}_{{}_{max}}:S^{n}\to\mathbb{R}_{+}$ be the function defined as $\widetilde{\gamma}_{{}_{max}}(\theta)={\rm max}\{\gamma_{{}_{1}}(\theta),\dots,\gamma_{{}_{m}}(\theta)\}.$ Then the Wulff shape of $\widetilde{\gamma}_{{}_{max}}$ is the convex hull of $\mathcal{W}_{\gamma_{{}_{1}}}\cup\dots\cup\mathcal{W}_{\gamma_{{}_{m}}}$.

4. Proof of Theorem 2

We first prove that the equality $\mathcal{W}_{\gamma_{{}_{min}}}=\mathcal{W}_{\gamma_{{}_{1}}}\cap\mathcal{W}_{\gamma_{{}_{2}}}$ holds for any convex integrands $\gamma_{{}_{1}},\gamma_{{}_{2}}$.

Lemma 4.1**.**

Let $\bar{\gamma}_{{}_{1}},\bar{\gamma}_{{}_{2}}$ be convex integrands such that $\mathcal{W}_{\gamma_{{}_{1}}}=\mathcal{W}_{\bar{\gamma}_{{}_{1}}},\mathcal{W}_{\gamma_{{}_{2}}}=\mathcal{W}_{\bar{\gamma}_{{}_{2}}}$. Set $\bar{\gamma}_{{}_{min}}:S^{n}\to\mathbb{R}_{+}$ is the continuous function defined by $\bar{\gamma}_{{}_{min}}(\theta)={\rm min}\{\bar{\gamma}_{{}_{1}}(\theta),\bar{\gamma}_{{}_{2}}(\theta)\}.$ Then $\mathcal{W}_{\bar{\gamma}_{{}_{min}}}=\mathcal{W}_{\bar{\gamma}_{{}_{1}}}\cap\mathcal{W}_{\bar{\gamma}_{{}_{2}}}$.

Proof of Lemma 4.1. By Proposition 2, $\mathcal{W}_{\bar{\gamma}_{{}_{min}}}$ can be rewritten as

[TABLE]

Here, the second equality follows from Proposition 5 and the thrid equality follows from Maehara’s lemma. Thus it is sufficient to prove the following:

[TABLE]

First, we show that

[TABLE]

Let $\widetilde{P}\in(\mathcal{D}\widetilde{\mathcal{W}}_{\bar{\gamma}_{{}_{1}}}\cup\mathcal{D}\widetilde{\mathcal{W}}_{\bar{\gamma}_{{}_{2}}})^{\circ}$. Then it follows that $\mathcal{D}\widetilde{\mathcal{W}}_{\bar{\gamma}_{{}_{1}}}\subset H(\widetilde{P})$. By Proposition 3, we have

[TABLE]

In the same way, we have that $\widetilde{P}\in\widetilde{\mathcal{W}}_{\bar{\gamma}_{{}_{2}}}$. Thus $\widetilde{P}\in\widetilde{\mathcal{W}}_{\bar{\gamma}_{{}_{1}}}\cap\widetilde{\mathcal{W}}_{\bar{\gamma}_{{}_{2}}}$.

Next, we show that

[TABLE]

Let $\widetilde{P}$ be a point of $\widetilde{\mathcal{W}}_{\bar{\gamma}_{{}_{1}}}\cap\widetilde{\mathcal{W}}_{\bar{\gamma}_{{}_{2}}}$. Since $\widetilde{\mathcal{W}}_{\bar{\gamma}_{{}_{1}}}$ and $\widetilde{\mathcal{W}}_{\bar{\gamma}_{{}_{2}}}$ are spherical convex bodies, by Proposition 1, it follows that

[TABLE]

where $i=1,2$. This implies

[TABLE]

Then it follows that $\mathcal{D}\widetilde{\mathcal{W}}_{\bar{\gamma}_{{}_{1}}}\cup\mathcal{D}\widetilde{\mathcal{W}}_{\bar{\gamma}_{{}_{2}}}$ is a subset of $H(\widetilde{P})$. By the definition of spherical polar set, $\widetilde{P}$ is a point of $(\mathcal{D}\widetilde{\mathcal{W}}_{\bar{\gamma}_{{}_{1}}}\cup\mathcal{D}\widetilde{\mathcal{W}}_{\bar{\gamma}_{{}_{2}}})^{\circ}$. Therefore, $\widetilde{\mathcal{W}}_{\bar{\gamma}_{{}_{1}}}\cap\widetilde{\mathcal{W}}_{\bar{\gamma}_{{}_{2}}}$ is a subset of $(\mathcal{D}\widetilde{\mathcal{W}}_{\bar{\gamma}_{{}_{1}}}\cup\mathcal{D}\widetilde{\mathcal{W}}_{\bar{\gamma}_{{}_{2}}})^{\circ}$. $\square$

The next lemma is useful in the coming proof.

Lemma 4.2**.**

The following inclusion is holds.

[TABLE]

Proof of Lemma 4.2. Since $\gamma_{{}_{min}}(\theta)\leq\gamma_{{}_{1}}(\theta)$, for any $\theta\in S^{n}$, it follows that

[TABLE]

Set

[TABLE]

By the properties $(1)-(3)$ of spherical blow-up $\Psi_{N}$, we have

[TABLE]

for any $\theta\in S^{n}$ (see Figure 6).

Thus it follows that

[TABLE]

Since $\bar{\gamma}_{{}_{1}}$ is the convex integrand of $\mathcal{W}_{\gamma_{{}_{1}}}$, by Proposition 3, we have

[TABLE]

Putting ($1$) together with ($2$), we conclude that

[TABLE]

In the same way, it follows that

[TABLE]

Because $\Psi_{N}\circ\alpha_{{}_{N}}^{-1}\circ Id({\rm graph}(\bar{\gamma}_{{}_{i}}))$ is a subset of $\mbox{s-conv}\left(\Psi_{N}\circ\alpha_{{}_{N}}^{-1}\circ Id({\rm graph}(\bar{\gamma}_{{}_{i}}))\right)$, $i=1,2,$ it follows that

[TABLE]

Here, the equality follows from definition of $\bar{\gamma}_{{}_{min}}$, the last inclusion follows from ($3$), ($4$) and

[TABLE]

is the smallest spherical convex set containing

[TABLE]

Then by Lemma 2.2, we conclude that

[TABLE]

$\square$

We are now in the position to show the equality \mathcal{W}_{\gamma_{{}_{min}}}=\bigl{(}\mathcal{W}_{\gamma_{{}_{1}}}\cap\mathcal{W}_{\gamma_{{}_{2}}}\bigr{)} holds for any support functions $\gamma_{{}_{1}},\gamma_{{}_{2}}$.

By Lemma 4.1, it is sufficient to prove that

[TABLE]

Since $\bar{\gamma}_{{}_{1}}(\theta)\leq\gamma_{{}_{1}}(\theta)$ and $\bar{\gamma}_{{}_{2}}(\theta)\leq\gamma_{{}_{2}}(\theta)$ for any $\theta\in S^{n}$, we have taht

[TABLE]

This implies

[TABLE]

So it is sufficient to prove that

[TABLE]

By Proposition 2, this follows from following:

[TABLE]

Here, the second and forth equalities follows from Maehara’s lemma, the inclusion follows from Lemma 4.2. This proves the Theorem. $\square$

Corollary 2**.**

Let $\gamma_{{}_{1}},\dots,\gamma_{{}_{m}}$ be support functions. Let $\widetilde{\gamma}_{{}_{min}}:S^{n}\to\mathbb{R}_{+}$ be the function defined as $\widetilde{\gamma}_{{}_{min}}(\theta)={\rm min}\{\gamma_{{}_{1}}(\theta),\dots,\gamma_{{}_{m}}(\theta)\}.$ Then the Wulff shape of $\widetilde{\gamma}_{{}_{min}}$ is the intersection $\mathcal{W}_{\gamma_{{}_{1}}}\cap\dots\cap\mathcal{W}_{\gamma_{{}_{m}}}$.

5. More topics on maximum and minimum of convex integrands

For given convex integrands (resp. continuous functions) $\gamma_{{}_{1}}$ and $\gamma_{{}_{2}}$, by proof of Theorem 1 (resp. Theorem 2), we have the following relation between $\mathcal{DW}_{\gamma_{{}_{1}}}$, $\mathcal{DW}_{\gamma_{{}_{2}}}$ and $\mathcal{W}_{\gamma_{{}_{max}}}$ (resp. $\mathcal{DW}_{\gamma_{{}_{1}}}$, $\mathcal{DW}_{\gamma_{{}_{2}}}$ and $\mathcal{W}_{\gamma_{{}_{min}}}$):

[TABLE]

[TABLE]

Since $\mathcal{DW}_{\gamma_{{}_{1}}}\cap\mathcal{DW}_{\gamma_{{}_{2}}}$ is a subset of the $\mbox{convex hull of }\mathcal{DW}_{\gamma_{{}_{1}}}\cup\mathcal{DW}_{\gamma_{{}_{2}}}$, by Lemma 2.2, it follows that

[TABLE]

Moreover, as a corollary, we have the following:

Corollary 3**.**

Let $\gamma_{{}_{1}}$ and $\gamma_{{}_{2}}$ be convex integrands. Suppose that $\mathcal{W}_{\gamma_{{}_{1}}}$ is the dual Wulff shape of $\mathcal{W}_{\gamma_{{}_{2}}}$. Then $\mathcal{W}_{\gamma_{{}_{max}}}$ is the dual Wulff shape of $\mathcal{W}_{\gamma_{{}_{min}}}$.

Proof.

By Theorem 1 and 2, we know that

[TABLE]

Since Maehara’s lemma implies

[TABLE]

it is sufficient to prove that

[TABLE]

Let $\widetilde{P}$ be a point of $(\widetilde{W}_{\gamma_{{}_{1}}}\cup\widetilde{W}_{\gamma_{{}_{2}}})^{\circ}$. Then it follows that

[TABLE]

This implies $\widetilde{P}$ is a point of $\widetilde{W}_{\gamma_{{}_{1}}}^{\circ}\cap\widetilde{W}_{\gamma_{{}_{2}}}^{\circ}$. Since $\widetilde{W}_{\gamma_{{}_{1}}}$ is the dual of $\widetilde{W}_{\gamma_{{}_{2}}}$, namely,

[TABLE]

it follows that

[TABLE]

Therefore, we conclude that

[TABLE]

On the other hand, since Wulff shapes $\mathcal{W}_{\gamma_{{}_{1}}},\mathcal{W}_{\gamma_{{}_{2}}}$ are duals, we have

[TABLE]

Let \widetilde{P}\in\bigl{(}\widetilde{W}_{\gamma_{{}_{1}}}\cap\widetilde{W}_{\gamma_{{}_{2}}}\bigr{)}. By ($5$), it follows that

[TABLE]

Then we have

[TABLE]

Therefore, it follows that

[TABLE]

∎

Acknowledgements

The author would like to express his sincere appreciation to Takashi Nishimura, for his kind advice. This work was partially supported by Natural Science Basic Research Program of Shaanxi (Program No. 2020JQ-235) and the Initial Foundation for Scientific Research of Northwest A&F University (Program No. 2452018018).