Random Polytopes and Affine Surface Area

Let K be a convex body in R. A random polytope is the convex hull [x1, ..., xn] of finitely many points chosen at random in K. 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 C3 boundary and everywhere positive curvature c(d) lim n→∞ vold(K)− E(K,n) ( vold(K) n ) 2 d+1 = ∫ ∂K κ(x) 1 d+1 dμ(x) where κ(x) denotes the Gauß-Kronecker curvature. We show that the same formula holds for all convex bodies if κ(x) denotes the generalized Gauß-Kronecker curvature. 1991 Mathematics Subject Classification. 52A22. Supported by NSF-grant DMS-9301506 Typeset by AMS-TEX