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 e-prints
- Pub Date:
- May 2014
- DOI:
- 10.48550/arXiv.1405.2503
- arXiv:
- arXiv:1405.2503
- Bibcode:
- 2014arXiv1405.2503J
- Keywords:
-
- Mathematics - Combinatorics;
- 52C35;
- 52C45;
- 68U05
- E-Print:
- 8 pages, 5 figures, published, Electron. J. Combin., corrections suggested by the referee have been incorporated