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A Note on Random Triangles
Author(s) -
Halász Sylvia,
Kleitman D. J.
Publication year - 1974
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1974533225
Subject(s) - mathematics , combinatorics , conjecture , simplex , product (mathematics) , center (category theory) , symmetry (geometry) , random variable , point (geometry) , simple (philosophy) , real line , discrete mathematics , geometry , statistics , philosophy , chemistry , epistemology , crystallography
In this paper we prove that the point in a triangle T maximizing the probability that it lies within a randomly chosen triangle within T is the center of T . A simple general closed expression for the indicated probability at any point P of T is obtained, though the proof proceeds by a symmetry argument on contributions to derivatives of the probability. It is also shown that the indicated probability is linearly related to the average over points in T of the product of the areas within T on either side of the line determined by P and . This gives the proof for n = 3 of the following conjecture of A. Prékopa: Fix a point P in an n − 1 dimensional simplex S . Take n independent points, uniformly distributed in S , and consider the probability ρ (P) of P being inside the random simplex determined by these n points. Conjecture: ρ (P) is maximal, when P is the center of gravity of S .