Random Points in Isotropic Unconditional Convex Bodies

Abstract

The paper considers three questions about independent random points uniformly distributed in isotropic symmetric convex bodies K, T 1 , .

Sampling convex bodies: a random matrix approach

“…Gathering the V-graphs with the same , we get as a consequence of inequality (10) and Lemma 3(d) that (setting m = l − 1)…”

“…When plugged in Rudelson's inequality, it yields that if N C(ε)n log n, we have A N (X) − Id ε with probability larger than 1 − ε (see [10,19]). On the other hand, when X is isotropic we have E|X| 2 = n and consequently we must take N larger than cn log n to use Rudelson's inequality.…”

Concentration of mass on convex bodies

“…See also [20] for an extension of this result. In [18] it was observed that N 0 c(ε)n log n is enough for the class of unconditional isotropic convex bodies. Theorem 1.1 allows us to prove the same fact in full generality.…”

“…where we have used the fact that Following Rudelson's argument (see also [18], page 10) we see that if x 1 , . .…”

“…For the proof of Theorem 1.6 we follow the argument of [18] which incorporates the concentration estimate of Theorem 1.1 into Rudelson's approach to the problem. The main lemma in [36] is the following.…”