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The Exact Distribution of the Number of Vertices of a Random Convex Chain
Author(s) -
Buchta Christian
Publication year - 2006
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300000127
Subject(s) - mathematics , combinatorics , convex hull , regular polygon , convex body , distribution (mathematics) , chain (unit) , convex combination , geometry , mathematical analysis , convex optimization , physics , astronomy
Assume that n points P 1 ,…, P n are distributed independently and uniformly in the triangle with vertices (0, 1), (0, 0), and (1, 0). Consider the convex hull of (0, 1), P 1 ,…, P n , and (1, 0). The vertices of the convex hull form a convex chain. Letp k ( n )be the probability that the convex chain consists – apart from the points (0, 1) and (1, 0) – of exactly k of the points P 1 ,…, P n . Bárány, Rote, Steiger, and Zhang [ 3 ] proved thatp n ( n ) = 2 n / [ n ! ( n + 1 ) ! ] . The values ofp k ( n )are determined for k = 1,…, n − 1, and thus the distribution of the number of vertices of a random convex chain is obtained. Knowing this distribution provides the key to the answer of some long‐standing questions in geometrical probability.