RANDOM POINTS IN ISOTROPIC UNCONDITIONAL CONVEX BODIES

The paper considers three questions about independent random points uniformly distributed in isotropic symmetric convex bodies K, T1, . . . , Ts. (a) Let ε ∈ (0, 1) and let x1, . . . , xN be chosen from K. Is it true that if N C(ε)n log n, then I − 1 NL2 K N i=1 xi ⊗ xi < ε with probability greater than 1−ε? (b) Let xi be chosen from Ti . Is it true that the unconditional norm t = T 1 . . . T s s i=1 tixi K dxs . . . dx1 is well comparable to the Euclidean norm in Rs? (c) Let x1, . . . , xN be chosen from K. Let E (K,N) := E |conv{x1, . . . , xN }|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.