Volume thresholds for Gaussian and spherical random polytopes and their duals

Peter Pivovarov¹

¹ Department of Mathematical and Statistical Sciences University of Alberta Edmonton, Alberta, Canada T6G 2G1

Studia Mathematica, Tome 183 (2007) no. 1, pp. 15-34

Résumé

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.

Détail

DOI : 10.4064/sm183-1-2

Keywords: gaussian random vector mathbb positive integer convex hull independent copies fix consider ratio volumes mathbb mathop vol cap mathop vol large range establish sharp threshold above which rightarrow rightarrow infty below which rightarrow rightarrow infty consider generated independent random vectors distributed uniformly euclidean sphere similar threshold results proved lastly prove complementary results polytopes generated random facets