## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 43, Number 1 (1972), 260-272.

### 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 $R(\mathbf{n})$.

#### Article information

**Source**

Ann. Math. Statist., Volume 43, Number 1 (1972), 260-272.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177692719

**Digital Object Identifier**

doi:10.1214/aoms/1177692719

**Mathematical Reviews number (MathSciNet)**

MR303650

**Zentralblatt MATH identifier**

0255.62018

**JSTOR**

links.jstor.org

#### Citation

Dempster, A. P. A Class of Random Convex Polytopes. Ann. Math. Statist. 43 (1972), no. 1, 260--272. doi:10.1214/aoms/1177692719. https://projecteuclid.org/euclid.aoms/1177692719