On the convex hull of uniform random points in a simpled-polytope

Denote the expected number of facets and vertices and the expected volume of the convex hullP _n ofn random points, selected independently and uniformly from the interior of a simpled-polytope byE _n (f), E _n (v), andE _n (V), respectively. In this note we determine the sharp constants of the asymptotic expansion ofE _n (f), E _n (v), andE _n (V), asn tends to infinity. Further, some results concerning the expected shape ofP _n are given. The work of F. Affentranger was supported by the Swiss National Foundation.