On the isotropic constant of random polytopes

Let X1, . . . , XN be independent random vectors uniformly distributed on an isotropic convex body K ⊂ Rn, and let KN be the symmetric convex hull of Xi’s. We show that with high probability LKN ≤ C √ log(2N/n), where C is an absolute constant. This result closes the gap in known estimates in the range Cn ≤ N ≤ n1+δ. Furthermore, we extend our estimates to the symmetric convex hulls of vectors y1X1, . . . , yNXN , where y = (y1, . . . , yN ) is a vector in RN . Finally, we discuss the case of a random vector y. 2010 Classification: Primary 52A23, 46B06; Secondary: 52A22, 52B11, 60D05