A Slight Improvement to the Colored Bárány's Theorem

Abstract: Suppose $d+1$ absolutely continuous probability measures $m_0, \ldots, m_d$ on $\mathbb{R}^d$ are given. In this paper, we prove that there exists a point of $\mathbb{R}^d$ that belongs to the convex hull of $d+1$ points $v_0, \ldots, v_d$ with probability at least $\frac{2d}{(d+1)!(d+1)}$, where each point $v_i$ is sampled independently according to probability measure $m_i$.