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arXiv · 2304.13133

On the probability that convex hull of random points contains the origin

Konstantin Tikhomirov
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Abstract

The classical theorem of Wendel provides an exact formula for the probability that the convex hull of independent symmetrically distributed vectors in ${\mathbb R}^d$ contains the origin as long as the distributions of the vectors are continuous. In this note, we provide an extension to Wendel's theorem for independent random vectors $X_1,\dots,X_n$ with i.i.d components having a (possibly discrete) symmetric distribution of unit variance. As a related observation, we give sharp estimates on the probability that a random linear program of the form ``$\max\langle x,{\mathfrak c}\rangle\quad\mbox{subject to }\langle X_i,x\rangle\leq 1,\;i\leq n$'', is bounded.

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Konstantin Tikhomirov. 2023-04-25. On the probability that convex hull of random points contains the origin. https://arxiv.org/abs/2304.13133

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