© The London Mathematical Society
Random Points in Isotropic Unconditional Convex Bodies
Department of Mathematics, University of Athens Panepistimiopolis, 157 84 Athens, Greece apgiannop{at}math.uoa.gr
Department of Mathematics, University of Missouri Columbia, MO 65211-4100, USA mariann{at}math.missouri.edu
Department of Mathematics, University of the Aegean Karlovassi, 832 00 Samos, Greece atsol{at}aegean.gr
Received 24 June 2004. Revision received 17 February 2005.
The paper considers three questions about independent random points uniformly distributed in isotropic symmetric convex bodies K, T1,...,Ts. (a) Let 
(0,1) and let x1,..., xN be chosen from K. Is it true that if N 
C()n log n, then
 |
with probability greater than 1–? (b) Let xi be chosen from Ti. Is it true that the unconditional norm
 |
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.