On the isotropic constant of random polytopes

Abstract

Abstract Let K be an isotropic 1-unconditional convex body in ℝ n . For every N > n consider N independent random points x 1 , . . . , x N uniformly distributed in K . We prove that, with probability greater than 1 – C 1 exp(– cn ) if N ≥ c 1 n and greater than 1 – C 1 exp(– cn / log n ) if n < N < c 1 n , the random polytopes K N ≔ conv{± x 1 , . . . , ± x N } and S N ≔ conv{ x 1 , . . . , x N } have isotropic constant bounded by an absolute constant C > 0.