Stability results for the volume of random simplices

TL;DR

This paper establishes stability estimates for the volume of random simplices in convex bodies, showing how near-minimal or near-maximal expected volumes imply the shape is close to an ellipsoid or a triangle.

Contribution

It provides the first stability results for the expected volume of random simplices in convex bodies, extending known extremal properties.

Findings

Stability estimates for minimal expected volume near ellipsoids.

Stability estimates for maximal expected volume near triangles in 2D.

Quantitative bounds relating shape proximity to expected volume extremality.

Abstract

It is known that for a convex body K in R^d of volume one, the expected volume of random simplices in K is minimised if K is an ellipsoid, and for d = 2, maximised if K is a triangle. Here we provide corresponding stability estimates.