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Solving a Generalized Heron Problem by Means of Convex Analysis

Boris S. Mordukhovich, Nguyen Mau Nam and Juan Salinas Jr.
The American Mathematical Monthly
Vol. 119, No. 2 (February 2012), pp. 87-99
DOI: 10.4169/amer.math.monthly.119.02.087
Stable URL: http://www.jstor.org/stable/10.4169/amer.math.monthly.119.02.087
Page Count: 13
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Solving a Generalized Heron Problem by Means of Convex Analysis
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Abstract

Abstract The classical Heron problem states: on a given straight line in the plane, find a point C such that the sum of the distances from C to the given points A and B is minimal. This problem can be solved using standard geometry or differential calculus. In the light of modern convex analysis, we are able to investigate more general versions of this problem. In this paper we propose and solve the following problem: on a given nonempty closed convex subset of ℝs, find a point such that the sum of the distances from that point to n given nonempty closed convex subsets of ℝs is minimal.

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