Volume thresholds for Gaussian and spherical random polytopes and their duals
Volume 183 / 2007
Abstract
Let $g$ be a Gaussian random vector in $\mathbb{R}^n$. Let $N=N(n)$ be a positive integer and let $K_N$ be the convex hull of $N$ independent copies of $g$. Fix $R>0$ and consider the ratio of volumes $V_N:={\mathbb E}\mathop{\rm vol}(K_N\cap RB_2^n)/\!\mathop{\rm vol}(RB_2^n)$. For a large range of $R=R(n)$, we establish a sharp threshold for $N$, above which $V_N\rightarrow 1$ as $n\rightarrow \infty$, and below which $V_N\rightarrow 0$ as $n\rightarrow \infty$. We also consider the case when $K_N$ is generated by independent random vectors distributed uniformly on the Euclidean sphere. In this case, similar threshold results are proved for both $R\in(0,1)$ and $R=1$. Lastly, we prove complementary results for polytopes generated by random facets.
