Equiaffine inner parallel curves of a plane convex body and the convex hulls of randomly chosen points

Abstract

A general formula is proved, which relates the equiaffine inner parallel curves of a plane convex body and the probability that the convex hull of j independent random points is disjoint from the convex hull of k further independent random points. This formula is applied to improve some well-known results in geometric probability. For example, an estimate, which was established for a special case by L. C. G. Rogers, is obtained with the best possible bound, and an asymptotic formula due to A. Renyi and R.␣Sulanke is extended to an asymptotic expansion.