Approximations of convex bodies by measure-generated sets

Abstract

Given a Borel measure $$\mu $$ on $$\mathbb {R}^{n}$$ , we define a convex set by $$\begin{aligned} \mathrm{M}\left( \mu \right) =\bigcup _{\begin{array}{c} 0\le f\le 1, \int _{\mathbb {R}^{n}}f\,\mathrm{d}\mu =1 \end{array} }\left\{ \int _{\mathbb {R}^{n}}yf\left( y\right) \,\mathrm{d}\mu \left( y\right) \right\} , \end{aligned}$$ where the union is taken over $$\mu $$ -measurable functions $$f:\mathbb {R}^{n}\rightarrow \left[ 0,1\right] $$ such that $$\int _{\mathbb {R}^{n}}f\,\mathrm{d}\mu =1$$ and $$\int _{\mathbb {R}^{n}}yf\left( y\right) \,\mathrm{d}\mu \left( y\right) $$ exists. 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 prove that for any non-degenerate probability measure $$\mu $$ , one has the lower bound $$\begin{aligned} \int _{\mathbb {R}^{n}}\left\| x\right\| _{Z_{1}\left( \mu \right) }\,\mathrm{d}\mu \left( x\right) \ge c\sqrt{n}, \end{aligned}$$ where $$Z_{1}\left( \mu \right) $$ is the $$L_{1}$$ -centroid body of $$\mu $$ .