A Slight Improvement to the Colored Bárány's Theorem

Abstract

Suppose $d+1$ absolutely continuous probability measures $m_0,\dots ,m_d$ 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 $v_0,\dots ,v_d$ with probability at least $\frac{2d}{(d+1)!(d+1)}$, where each point $v_i$ is sampled independently according to probability measure $m_i$.

8 pages, 5 figures, published, Electron. J. Combin., corrections suggested by the referee have been incorporated

References in corpus (1)

• Stabbing simplices by points and flats