On the convex hull of random points in a polytope

Abstract

The convex hull of n points drawn independently from a uniform distribution on the interior of a d -dimensional polytope is investigated. It is shown that the expected number of vertices is O (log d– 1 n ) for any polytope, the expected number of vertices is Ω(log d– 1 n ) for any simple polytope, and the expected number of facets is O (log d– 1 n ) for any simple polytope. An algorithm is presented for constructing the convex hull of such sets of points in linear average time.