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

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.