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. $\mathbb{E}(K,n)$ is the expectation of the volume of a random polytope of n randomly chosen points. I. Bárány showed that we have for convex bodies with $C^3$ boundary and everywhere positive curvature $$c(d)\underset{n\to \infty }{\lim }\frac{vo{l}_d(K)-\mathbb{E}(K,n)}{(\frac{vo{l}_d(K)}{n}{)}^{\frac{2}{d+1}}}={\int }_{\partial K}\kappa (x{)}^{\frac{1}{d+1}}d\mu (x)$$ where $\kappa (x)$ denotes the Gauß-Kronecker curvature. We show that the same formula holds for all convex bodies if $\kappa (x)$ denotes the generalized Gauß-Kronecker curvature.