# 凸体的λ Entropy的Brunn-Minkowski不等式The Brunn-Minkowski Inequalities of λ Entropy of Convex Body

DOI: 10.12677/PM.2021.117153, PDF, HTML, XML, 下载: 19  浏览: 56

Abstract: In the paper Gauss Image Problem, Böröczky-Lutwak-Yang-Zhang-Zhao proposed the Gaussian image problem, and under the assumption that the Borel measure λ is absolutely continuous, they proved the existence and uniqueness of the solution of the Gaussian image problem. In this paper, we establish the Brunn-Minkowski type inequality of the λ entropy of convex body. As a corollary, we give another proof of the uniqueness of the Gaussian image problem. Note that even if the measure λ is not absolutely continuous, the Brunn-Minkowski inequality of the λ entropy still holds.

1. 引言

Brunn-Minkowski不等式始现于19世纪末期，它在如今蓬勃发展的凸几何分析中扮演着十分重要的角色。Brunn-Minkowski不等式表明对于 ${ℝ}^{n}$ 中的凸体 $K,L$ 它们的体积及其代数和具有下述关系：

$V{\left(K+L\right)}^{\frac{1}{n}}\ge V{\left(K\right)}^{\frac{1}{n}}+V{\left(L\right)}^{\frac{1}{n}}$, (1.1)

Aleksandrov在20世纪早期开创性的工作 [1] 表明在Brunn-Minkowski 类型的不等式和混合表面积测度的唯一性之间存在着等价关系。在这里，“唯一性”意味着：如果K和L具有相同的混合表面积测度 ${S}_{i}\left(K,\cdot \right)={S}_{i}\left(L,\cdot \right)$，其中 $1\le i\le n-1$ [2]，那么K必是L的平移。

$V{\left(K\stackrel{˜}{+}L\right)}^{\frac{1}{n}}\le V{\left(K\right)}^{\frac{1}{n}}+V{\left(L\right)}^{\frac{1}{n}}$ (1.2)

${\stackrel{˜}{W}}_{n-q}\left(K\right)=\frac{1}{n}{\int }_{{S}^{n-1}}{\rho }_{K}^{q}\left(u\right)\text{d}u$, $q\ne 0$,

$\stackrel{˜}{E}\left(K\right)=\frac{1}{n}{\int }_{{S}^{n-1}}\mathrm{log}{\rho }_{K}\left(u\right)\text{d}u$

Brunn-Minkowski类型的不等式与Minkowski问题存在紧密联系。然而，已知的关于径向和 $K\stackrel{˜}{+}L$ 的对偶Brunn-Minkowski型不等式(1.2)式在测度 ${\stackrel{˜}{C}}_{p,q}\left(K,\cdot \right)$ 的唯一性问题上没有帮助。在许多公开的学术讨论中，Huang-Lutwak-Yang-Zhang关于对偶Brunn-Minkowski不等式曾猜测：

${\stackrel{˜}{W}}_{n-q}{\left(K+L\right)}^{\frac{1}{q}}\ge {\stackrel{˜}{W}}_{n-q}{\left(K\right)}^{\frac{1}{q}}+{\stackrel{˜}{W}}_{n-q}{\left(L\right)}^{\frac{1}{q}}$ (1.3)

${\stackrel{˜}{W}}_{n-q}\left(\left(1-t\right)\cdot K{+}_{0}t\cdot L\right)\ge {\stackrel{˜}{W}}_{n-q}{\left(K\right)}^{1-t}\stackrel{˜}{W}{\left(L\right)}^{t}$ (1.4)

$\lambda$ 是定义在 ${S}^{n-1}$ 上的Lebesgue可测集类上的子测度(submeasure)。 $K\in {\mathcal{K}}_{o}{}^{n}$，对任意的Borel集 $\omega \subset {S}^{n-1}$$\lambda$ 关于K的高斯像测度 $\lambda \left(K,\cdot \right)$ 可以定义为

$\lambda \left(K,\omega \right)=\lambda \left({\alpha }_{K}\left(\omega \right)\right)$,

$\lambda \left(K,\cdot \right)=\mu$

$\lambda \left(K,\cdot \right)={C}_{0}\left(K,\cdot \right)$,

$\lambda$ 是某个凸体K的曲率测度 ${C}_{n-1}\left(K,\cdot \right)$ 时，逆高斯像测度(详细定义参见第2节)就是表面积测度，即

${\lambda }^{*}\left(K,\cdot \right)={S}_{n-1}\left(K,\cdot \right)$

${\stackrel{˜}{E}}_{\lambda }\left(K\right)={\int }_{{S}^{n-1}}\mathrm{log}{\rho }_{K}\left(v\right)\text{d}\lambda \left(v\right)$.

Borel测度 $\mu$ 是绝对连续的，指的是 $\mu$ 是关于球面Lebesgue测度绝对连续的。Böröczky-Lutwak-Yang-Zhang-Zhao [14] 在 $\mu$ 是绝对连续的假设下，证明了高斯像问题的解存在唯一性。

${\stackrel{˜}{E}}_{\lambda }\left(\left(1-t\right)\cdot K{+}_{0}t\cdot L\right)\ge \left(1-t\right){\stackrel{˜}{E}}_{\lambda }\left(K\right)+t{\stackrel{˜}{E}}_{\lambda }\left( L \right)$

$\lambda \left(K,\cdot \right)=\lambda \left(L,\cdot \right)$

2. 预备知识

${h}_{K}\left(u\right)=\mathrm{max}\left\{〈x,u〉:x\in K\right\}$.

$\alpha >0$$x\in {ℝ}^{n}\overline{)}\left\{0\right\}$，凸体K的径向函数 ${\rho }_{K}$ 的定义为

${\rho }_{K}\left(x\right)=\mathrm{max}\left\{\alpha :\alpha x\in K\right\}$.

${H}_{K}\left(v\right)=\left\{x\in {ℝ}^{n}:〈x,v〉={h}_{K}\left(v\right)\right\}$.

$K\in {\mathcal{K}}_{o}{}^{n}$$K\ne \varnothing$，若对于任意的 $x\in K$，线段 $\left[0,x\right]\subset K$，则称K是关于原点o的星集。当一个星集的径向函数为正的且连续时，称该星集为星体。 ${ℝ}^{n}$ 中所有包含原点的星体组成一个星体类，记为 ${\mathcal{S}}_{o}{}^{n}$

${C}^{+}\left({S}^{n-1}\right)$ 是定义在 ${S}^{n-1}$ 上的正向函数类。对每一个 $f\in {C}^{+}\left({S}^{n-1}\right)$，由f生成的Wulff形记为 $\left[f\right]$，其定义为

$\left[f\right]=\left\{x\in {ℝ}^{n}:〈x,v〉\le f\left(v\right),\forall v\in {S}^{n-1}\right\}$.

$\Omega \subset {S}^{n-1}$ 是一个不包含 ${S}^{n-1}$ 上任意闭半球面的闭集。若 $h:\Omega \to \left(0,\infty \right)$$f:\Omega \to ℝ$ 是连续的， $\delta >0$ 以及

$\mathrm{log}{h}_{t}\left(v\right)=\mathrm{log}h\left(v\right)+tf\left(v\right)+o\left(t,v\right)$,

$g:\Omega \to ℝ$ 是连续的， $\delta >0$，对每一个 $t\in \left(-\delta ,\delta \right)$${\rho }_{t}:\Omega \to \left(0,\infty \right)$ 是连续的，以及

$\mathrm{log}{\rho }_{t}\left(u\right)=\mathrm{log}\rho \left(u\right)+tg\left(u\right)+o\left(t,u\right)$,

$〈{\rho }_{t}〉=conv\left\{{\rho }_{t}\left(u\right)u:u\in {S}^{n-1}\right\}$,

$〈{\rho }_{t}〉$ 是由 $\left(\rho ,g\right)$ 生成的对数族的凸包。如果 $\rho$ 恰好是某个凸体的径向函数，就将 $〈{\rho }_{t}〉$ 记为 $〈K,g〉$，并称 $〈K,g〉$ 是由 $\left(K,g\right)$ 生成的对数族的凸包。

$\alpha \in \left(0,1\right)$，星体的 ${L}_{p}$ 对偶组合 $\left(1-\alpha \right)\cdot K{\stackrel{˜}{+}}_{p}\alpha \cdot L$ (参见 [14] )按如下方式定义

$\left(1-\alpha \right)\cdot K{\stackrel{˜}{+}}_{p}\alpha \cdot L=\left\{tu:0\le t\le {\left(\left(1-\alpha \right){\rho }_{K}^{p}\left(u\right)+\alpha {\rho }_{L}^{p}\left(u\right)\right)}^{\frac{1}{p}},u\in {S}^{n-1}\right\},p\ne 0$

${K}^{*}=\left\{x\in {ℝ}^{n}:〈x,y〉\le 1,\forall y\in K\right\}$,

${\rho }_{K}=\frac{1}{{h}_{{K}^{*}}}$, ${h}_{K}=\frac{1}{{\rho }_{{K}^{*}}}$ (2.1)

${K}^{**}=K$

$K=cL$

$\sigma \subset \partial K$ 的球面像的定义为

${\upsilon }_{K}\left(\sigma \right)=\left\{v\in {S}^{n-1}:x\in {H}_{K}\left(v\right),\exists x\in \sigma \right\}\subset {S}^{n-1}$

${r}_{K}:{S}^{n-1}\to \partial K$, ${r}_{K}\left(u\right)={\rho }_{K}\left(u\right)u\in \partial K$,

${\alpha }_{K}\left(\omega \right)={\upsilon }_{K}\left({r}_{K}\left(\omega \right)\right)\subset {S}^{n-1}$.

${\alpha }_{K}^{*}\left(\eta \right)=\left\{u\in {S}^{n-1}:{\alpha }_{K}\left(u\right)\cap \eta \ne \varnothing \right\}$.

$\lambda$ 是定义在 ${S}^{n-1}$ 的Lebesgue可测子集的子测度， $K\in {\mathcal{K}}_{o}{}^{n}$。对任意的Borel集 $\omega \subset {S}^{n-1}$$\lambda$ 关于K的高斯像测度 $\lambda \left(K,\cdot \right)$ 可以定义为

$\lambda \left(K,\omega \right)=\lambda \left({\alpha }_{K}\left(\omega \right)\right)$.

${\lambda }^{*}\left(K,\omega \right)=\lambda \left({\alpha }_{K}^{*}\left(\omega \right)\right)=\lambda \left({\alpha }_{{K}^{*}}\left(\omega \right)\right)$.

$\lambda \left(cK,\cdot \right)=\lambda \left(K,\cdot \right)$, ${\lambda }^{*}\left(cK,\cdot \right)={\lambda }^{*}\left(K,\cdot \right)$

${\lambda }^{*}\left(K,\cdot \right)=\lambda \left({K}^{*},\cdot \right)$ (2.2)

$K\in {\mathcal{K}}_{o}{}^{n}$， [14] 还给出了关于测度 $\lambda$ 的K的对数体积 ${\lambda }_{0}\left(K\right)$

${\lambda }_{0}\left(K\right)=\mathrm{exp}\left\{\frac{1}{|\lambda |}{\int }_{{S}^{n-1}}\mathrm{log}{\rho }_{K}\left(u\right)\text{d}\lambda \left(u\right)\right\}$.

${\stackrel{˜}{E}}_{\lambda }\left(K\right)=|\lambda |\mathrm{log}{\lambda }_{0}\left(K\right)$ (2.3)

$\frac{\text{d}}{\text{d}t}{\mathrm{log}{\lambda }_{0}\left({〈K,f,t〉}^{*}\right)|}_{t=0}=\frac{1}{|\lambda |}{\int }_{{S}^{n-1}}g\left(u\right)\text{d}\lambda \left(K,u\right)$

$\left[K,f\right]$ 是由 $\left(K,f\right)$ 生成的Walff形，则

$\frac{\text{d}}{\text{d}t}{\mathrm{log}{\lambda }_{0}\left(\left[K,f,t\right]\right)|}_{t=0}=\frac{1}{|\lambda |}{\int }_{{S}^{n-1}}f\left(u\right)\text{d}{\lambda }^{*}\left(K,u\right)$

$\underset{t\to 0}{\mathrm{lim}}\frac{{\stackrel{˜}{E}}_{\lambda }\left(\left(1-t\right)\cdot K{+}_{0}t\cdot L\right)-{\stackrel{˜}{E}}_{\lambda }\left(K\right)}{t}={\int }_{{S}^{n-1}}\mathrm{log}\frac{{h}_{L}\left(u\right)}{{h}_{K}\left(u\right)}\text{d}\lambda \left({K}^{*},v\right)$

$\left[K,f,t\right]=\left(1-t\right)\cdot K{+}_{0}t\cdot L$,

f是连续的，结合引理2.1，(2.2)式以及(2.3)式，立即得证。

3. 关于高斯像测度的 Brunn-Minkowski不等式

$\left(1-\alpha \right)\cdot K{\stackrel{˜}{+}}_{p}\alpha \cdot L\subset \left(1-\alpha \right)\cdot K{+}_{p}\alpha \cdot L$, (3.1)

$〈x,v〉\le {h}_{\left(1-\alpha \right)\cdot K{+}_{p}\alpha \cdot L}\left(v\right)$, $\forall x\in \partial \left(\left(1-\alpha \right)\cdot K{\stackrel{˜}{+}}_{p}\alpha \cdot L\right)$ (3.2)

$〈{\rho }_{K}\left(u\right)u,v〉\le {h}_{K}\left(v\right)$, $〈{\rho }_{L}\left(u\right)u,v〉\le {h}_{L}\left(v\right)$, $\forall v\in {S}^{n-1}$

${\left(\left(1-\alpha \right){\rho }_{K}^{p}\left(u\right)+\alpha {\rho }_{L}^{p}\left(u\right)\right)}^{\frac{1}{p}}〈u,v〉\le {\left(\left(1-\alpha \right){h}_{K}^{p}\left(u\right)+\alpha {h}_{L}^{p}\left(u\right)\right)}^{\frac{1}{p}}$,

$〈u,v〉<0$，则上面的不等式显然成立。由此可得，在 $p\ne 0$ 时，对任意的 $\left(1-\alpha \right)\cdot K{\stackrel{˜}{+}}_{p}\alpha \cdot L$ 的边界点，(3.2)成立。

${\rho }_{K}^{1-\alpha }\left(u\right){\rho }_{L}^{\alpha }\left(u\right)〈u,v〉\le {h}_{K}^{1-\alpha }\left(v\right){h}_{L}^{\alpha }\left(v\right)$,

$p=0$ 时，(3.2)成立。

${\left(\left(1-\alpha \right){\rho }_{K}^{p}\left(u\right)+\alpha {\rho }_{L}^{p}\left(u\right)\right)}^{\frac{1}{p}}〈u,v〉={\left(\left(1-\alpha \right){h}_{K}^{p}\left(v\right)+\alpha {h}_{L}^{p}\left(v\right)\right)}^{\frac{1}{p}}$,

${\rho }_{K}\left(u\right)〈u,v〉={h}_{K}\left(v\right)$，以及 ${\rho }_{L}\left(u\right)〈u,v〉={h}_{L}\left(v\right)$

${\stackrel{˜}{E}}_{\lambda }\left(\left(1-t\right)\cdot K{+}_{0}t\cdot L\right)\ge \left(1-t\right){\stackrel{˜}{E}}_{\lambda }\left(K\right)+t{\stackrel{˜}{E}}_{\lambda }\left(L\right)$,

$\begin{array}{c}{\stackrel{˜}{E}}_{\lambda }\left(\left(1-t\right)\cdot K{+}_{0}t\cdot L\right)={\int }_{{S}^{n-1}}\mathrm{log}{\rho }_{\left(1-t\right)\cdot K{+}_{0}t\cdot L}\left(u\right)\text{d}\lambda \left(u\right)\\ \ge {\int }_{{S}^{n-1}}\mathrm{log}{\rho }_{\left(1-t\right)\cdot K{\stackrel{˜}{+}}_{0}t\cdot L}\left(u\right)\text{d}\lambda \left(u\right)\\ ={\int }_{{S}^{n-1}}\mathrm{log}{\rho }_{K}^{1-t}\left(u\right){\rho }_{L}^{t}\left(u\right)\text{d}\lambda \left(u\right)\\ =\left(1-t\right){\stackrel{˜}{E}}_{\lambda }\left(K\right)+t{\stackrel{˜}{E}}_{\lambda }\left( L \right)\end{array}$

4. 高斯像测度的唯一性

$\lambda \left(K,\cdot \right)=\lambda \left(L,\cdot \right)$

${F}^{\prime }\left(0\right)\ge F\left(1\right)-F\left(0\right)$.

$\lambda \left(K,\cdot \right)=\lambda \left(L,\cdot \right)$ 时，如果K和L不是膨胀的，我们就得到严格的不等式

${F}^{\prime }\left(0\right)>F\left(1\right)-F\left(0\right)$.

${\int }_{{S}^{n-1}}\mathrm{log}\frac{{h}_{{L}^{*}}\left(v\right)}{{h}_{{K}^{*}}\left(v\right)}\text{d}\lambda \left(K,v\right)>{\stackrel{˜}{E}}_{\lambda }\left({L}^{*}\right)-{\stackrel{˜}{E}}_{\lambda }\left({K}^{*}\right)$ (4.1)

${\int }_{{S}^{n-1}}\mathrm{log}\frac{{h}_{{K}^{*}}\left(v\right)}{{h}_{{L}^{*}}\left(v\right)}\text{d}\lambda \left(L,v\right)>{\stackrel{˜}{E}}_{\lambda }\left({K}^{*}\right)-{\stackrel{˜}{E}}_{\lambda }\left({L}^{*}\right)$ (4.2)

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