arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1706.07112math.MG
收藏

凸体的测度生成集逼近

Approximations of convex bodies by measure-generated sets

Han Huang, Boaz A. Slomka

备注：
24 pages, 1 figure
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

给定${\mathbb R}^{n}$上的Borel测度$μ$，我们用\[ M(μ)=\bigcup_{\substack{0\le f\le1,\\ \int_{{\mathbb R}^{n}}f\,{\rm d}μ=1 } }\left\{ \int_{{\mathbb R}^{n}}yf\left(y\right)\,{\rm d}μ\left(y\right)\right\} , \]定义了一个凸集，这里的并集是所有$μ$ -可测函数$f:{\mathbb R}^{n}\to\left[0,1\right]$与$\int_{{\mathbb R}^{n}}f\,{\rm d}μ=1$的并集.我们研究了这些测度生成集的性质，并利用它们来研究一般凸体用尽可能少顶点的多面体逼近问题的自然变化。特别地，我们研究了Bezdek和Litvak引入的顶点指数的扩展。作为应用，我们给出了非退化概率测度的质心体的某些平均范数的下界。

英文摘要：

Given a Borel measure $μ$ on ${\mathbb R}^{n}$, we define a convex set by \[ M(μ)=\bigcup_{\substack{0\le f\le1,\\ \int_{{\mathbb R}^{n}}f\,{\rm d}μ=1 } }\left\{ \int_{{\mathbb R}^{n}}yf\left(y\right)\,{\rm d}μ\left(y\right)\right\} , \] where the union is taken over all $μ$-measurable functions $f:{\mathbb R}^{n}\to\left[0,1\right]$ with $\int_{{\mathbb R}^{n}}f\,{\rm d}μ=1$. We study the properties of these measure-generated sets, and use them to investigate natural variations of problems of approximation of general convex bodies by polytopes with as few vertices as possible. In particular, we study an extension of the vertex index which was introduced by Bezdek and Litvak. As an application, we provide a lower bound for certain average norms of centroid bodies of non-degenerate probability measures.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑