A Class of Random Convex Polytopes

Abstract

In Section 1 it is shown that $n$ interior points of the $(k - 1)$-dimensional simplex $S_k$ define a partition of $S_k$ into $\binom{n+k-1}{k-1}$ convex polytopes $R(\mathbf{n})$ which are in one-one correspondence with the partitions of $n$ into a sum of $k$ nonnegative integers. If the $n$ points are uniformly and independently distributed over $S_k$, then $R(\mathbf{n})$ becomes a random polytope. Basic properties of the random $R(\mathbf{n})$ are given in Section 2. Section 3 presents an algorithm which can be used to compute the distribution of any extremal vertex of $R(\mathbf{n})$.