%0 Journal Article
%T Intrinsic volumes of inscribed random polytopes in smooth convex bodies
%A Imre B¨¢r¨¢ny
%A Ferenc Fodor
%A Viktor V¨ªgh
%J Mathematics
%D 2009
%I arXiv
%R 10.1239/aap/1282924055
%X Let $K$ be a $d$ dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by $K_n$ the convex hull of $n$ points chosen randomly and independently from $K$ according to the uniform distribution. Matching lower and upper bounds are obtained for the orders of magnitude of the variances of the $s$-th intrinsic volumes $V_s(K_n)$ of $K_n$ for $s\in\{1, ..., d\}$. Furthermore, strong laws of large numbers are proved for the intrinsic volumes of $K_n$. The essential tools are the Economic Cap Covering Theorem of B\'ar\'any and Larman, and the Efron-Stein jackknife inequality.
%U http://arxiv.org/abs/0906.0309v1