On symmetric versions of Sylvester's problem

TL;DR

This paper studies the moments of the normalized volume of symmetric and nonsymmetric random polytopes within convex bodies, identifying extremal cases and calculating exact values, with a focus on planar parallelograms.

Contribution

It characterizes extremal moments of random polytopes in convex bodies and provides exact calculations for specific cases, advancing understanding of geometric probability.

Findings

Moments are maximized by parallelograms among planar convex bodies.

Exact values of moments are computed in certain extremal cases.

Symmetric and nonsymmetric cases are both analyzed.

Abstract

We consider moments of the normalized volume of a symmetric or nonsymmetric random polytope in a fixed symmetric convex body. We investigate for which bodies these moments are extremized, and calculate exact values in some of the extreme cases. We show that these moments are maximized among planar convex bodies by parallelograms.