A Slight Improvement to the Colored Bárány's Theorem

Zilin Jiang
Suppose d + 1 absolutely continuous probability measures m 0 ,. .. , m d on R d are given. In this paper, we prove that there exists a point of R d that belongs to the convex hull of d + 1 points v 0 ,. .. , v d with probability at least 2d (d+1)!(d+1) , where each point v i is sampled independently according to probability measure m i .