# Approximations of convex bodies by measure-generated sets

## TL;DR

This paper introduces measure-generated convex sets and explores their properties to improve approximation of convex bodies by polytopes, extending the vertex index and providing bounds for centroid bodies of probability measures.

## Contribution

It defines and analyzes measure-generated convex sets and applies these to enhance convex body approximation and extend the vertex index concept.

## Key findings

- Properties of measure-generated convex sets are characterized.
- New bounds are established for approximation of convex bodies.
- An extension of the vertex index is proposed and analyzed.

## Abstract

Given a Borel measure $\mu$ on ${\mathbb R}^{n}$, we define a convex set by \[ M({\mu})=\bigcup_{\substack{0\le f\le1,\\ \int_{{\mathbb R}^{n}}f\,{\rm d}{\mu}=1 } }\left\{ \int_{{\mathbb R}^{n}}yf\left(y\right)\,{\rm d}{\mu}\left(y\right)\right\} , \] where the union is taken over all $\mu$-measurable functions $f:{\mathbb R}^{n}\to\left[0,1\right]$ with $\int_{{\mathbb R}^{n}}f\,{\rm d}{\mu}=1$. 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.