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February, 1972 A Class of Random Convex Polytopes
A. P. Dempster
Ann. Math. Statist. 43(1): 260-272 (February, 1972). DOI: 10.1214/aoms/1177692719


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 $R(\mathbf{n})$.


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A. P. Dempster. "A Class of Random Convex Polytopes." Ann. Math. Statist. 43 (1) 260 - 272, February, 1972.


Published: February, 1972
First available in Project Euclid: 27 April 2007

MathSciNet: MR303650
zbMATH: 0255.62018
Digital Object Identifier: 10.1214/aoms/1177692719

Rights: Copyright © 1972 Institute of Mathematical Statistics


Vol.43 • No. 1 • February, 1972
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