On the isotropy constant of random convex sets

Abstract

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