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 $m_i$.
 Publication:

arXiv eprints
 Pub Date:
 May 2014
 arXiv:
 arXiv:1405.2503
 Bibcode:
 2014arXiv1405.2503J
 Keywords:

 Mathematics  Combinatorics;
 52C35;
 52C45;
 68U05
 EPrint:
 8 pages, 5 figures, published, Electron. J. Combin., corrections suggested by the referee have been incorporated