# A Slight Improvement to the Colored Bárány's Theorem

@article{Jiang2014ASI, title={A Slight Improvement to the Colored B{\'a}r{\'a}ny's Theorem}, author={Zilin Jiang}, journal={Electr. J. Comb.}, year={2014}, volume={21}, pages={P4.39} }

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 $m_i$.

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## Positive-fraction intersection results and variations of weak epsilon-nets

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