Estimating Support Functions of Random Polytopes via Orlicz Norms

TL;DR

This paper introduces a novel probabilistic approach using Orlicz norms to estimate the expected support functions of random polytopes generated from uniform distributions in isotropic convex bodies.

Contribution

It presents a new method leveraging Orlicz norms and probabilistic estimates to analyze support functions of random polytopes, a connection not previously explored.

Findings

New bounds for support functions of random polytopes

Application of Orlicz norms in convex geometry

Enhanced understanding of random polytope structure

Abstract

We study the expected value of support functions of random polytopes in a certain direction, where the random polytope is given by independent random vectors uniformly distributed in an isotropic convex body. All results are obtained by an utterly novel approach, using probabilistic estimates in connection with Orlicz norms that were not used in this connection before.