EQUIAFFINE INNER PARALLEL CURVES OF A PLANE CONVEX BODY AND THE CONVEX HULLS OF RANDOMLY CHOSEN POINTS

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