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

We show that for every $K\geq 1$ there is $c>0$ depending only on $K$ with the following property. Let $n>d\geq 1$, and let $X_1,\dots,X_n$ be independent random vectors with i.i.d components having a (possibly discrete) symmetric distribution of unit variance and the subgaussian moment bounded by $K$. Set $p_{n,d}:=1-2^{-n+1}\sum_{k=0}^{d-1}{n-1\choose k}.$ Then \begin{align*} p_{n,d}\leq {\mathbb P}\big\{{\rm conv}\{X_1,\dots,X_n\}\mbox{ contains the origin}\big\} \leq p_{n,d} +2\exp(-cd), \end{align*} and \begin{align*} p_{n,d}-2\exp(-cd)\leq {\mathbb P}\big\{{\rm conv}\{X_1,\dots,X_n\}\mbox{ contains the origin in the interior}\big\} \leq p_{n,d}. \end{align*} We further prove a related result in the context of average-case analysis of linear programs. Let $n\geq d$, let ${\bf 1}$ be an $n$-dimensional vector of all ones, and let $A$ be an $n\times d$ random matrix with i.i.d symmetrically distributed entries of unit variance and subgaussian moment bounded above by $K$. Then for any non-zero non-random cost vector $\mathfrak c$, $$ \big| {\mathbb P}\big\{\mbox{Linear program ``$\max\langle x,{\mathfrak c}\rangle\quad\mbox{subject to }Ax\leq {\bf 1}$'' is bounded}\big\} -p_{n+1,d} \big|\leq 2\exp(-cd). $$ In particular, the result implies that for $n=2d+\omega(\sqrt{d})$, the linear program is bounded with probability $1-o(1)$, and for $n=2d-\omega(\sqrt{d})$ unbounded with probability $1-o(1)$.