Random Points in Isotropic Unconditional Convex Bodies

The paper considers three questions about independent random points uniformly distributed in isotropic symmetric convex bodies K, T 1 ,…, T s . (a) Let ɛ ∈ (0,1) and let x 1 ,…, x N be chosen from K . Is it true that if N ⩾ C (ɛ) n log n , then‖ I − 1 N L K 2∑ i = 1 Nx i ⊗ x i‖ < εwith probability greater than 1−ɛ? (b) Let x i be chosen from T i . Is it true that the unconditional norm‖ t ‖ = ∫ T 1… ∫ T s‖∑ i = 1 st i x i‖ K d x s … d x 1is well comparable to the Euclidean norm in R s ? (c) Let x 1 ,…, x N be chosen from K . Let E( K,N ):=E| conv { x 1 ,…, x N }| 1/ n be the expected volume radius of their convex hull. Is it true that E( K,N )≃ E( B ( n ), N ) for all N , where B ( n ) is the Euclidean ball of volume 1? It is proved that the answers to these questions are affirmative if there is a restriction to the class of unconditional convex bodies. The main tools come from recent work of Bobkov and Nazarov. Some observations about the general case are also included.