About the isotropy constant of random convex sets

TL;DR

This paper proves that the isotropy constant of a symmetric convex hull of sufficiently many random vectors on the sphere is bounded with high probability, advancing understanding of geometric properties of random convex sets.

Contribution

It establishes a probabilistic bound on the isotropy constant for convex hulls of random vectors, extending previous results to a broader range of m relative to n.

Findings

Isotropy constant is bounded with high probability when m ≥ (1+δ)n.

The bound depends only on δ, not on n or m directly.

Results hold for vectors uniformly distributed on the sphere.

Abstract

Let K be the symmetric convex hull of m independent random vectors uniformly distributed on the unit sphere of R^n. We prove that, for every $\delta>0$, the isotropy constant of K is bounded by a constant $c(\delta)$ with high probability, provided that $m\geq (1+\delta)n$.