Random polytopes and affine surface area

Abstract

Let K be a convex body in $R^d$. A random polytope is the convex hull $[x_1,...,x_n]$ of finitely many points chosen at random in K. $\Bbb E(K,n)$ is the expectation of the volume of a random polytope of n randomly chosen points. I. B\'ar\'any showed that we have for convex bodies with $C^3$ boundary and everywhere positive curvature $$ c(d)\lim_{n \to \infty} \frac {vol_d(K)-\Bbb E(K,n)}{(\frac{vol_d(K)}{n})^{\frac{2}{d+1}}} =\int_{\partial K} \kappa(x)^{\frac{1}{d+1}}d\mu(x) $$ where $\kappa(x)$ denotes the Gau\ss-Kronecker curvature. We show that the same formula holds for all convex bodies if $\kappa(x)$ denotes the generalized Gau\ss-Kronecker curvature.