Solving a Generalized Heron Problem by Means of Convex Analysis

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 IR!, find a point such that the sum of the distances from that point to n given nonempty closed convex subsets of JR• is minimal. 1 Problem Formulation. Heron from Alexandria (10-75 AD) was "a Greek geometer and inventor whose writings preserved for posterity a knowledge of the mathematics and engineering of Babylonia, ancient Egypt, and the Greco-Roman world" (from the Encyclopedia Britannica). One of the geometric problems he proposed in his Catroptica was as follows: find a point on a straight line in the plane such that the sum of the distances from it to two given points is minimal. Recall that a subset n of JRS is called convex if AX+ (1 .>.)y E n whenever X and y are in n and 0 :::; .>. :::; 1. Our idea now is to consider a much broader situation, where two given points in the classical Heron problem are replaced by finitely many closed and convex subsets ni, i = 1, ... , n, and the given line is replaced by a given closed and convex subset n of JRS 0 We are looking for a point on the set n such that the sum of the distances from that point to ni, i = 1, 0 0 0 'n, is minimal. The distance from a point x to a nonempty set n is understood in the conventional way d(x; D)= inf {llxYll\ y ED}, (1.1) where II · II is the Euclidean norm in lR • The new generalized Heron problem is formulated as follows: n minimize D(x) := L d(x; ni) subject to X En, (1.2) i=l where all the sets n and ni, i = 1, ... , n, are nonempty, closed, and convex; these are our standing assumptions in this paper. Thus (1.2) is a constrained convex optimization 1 Department of Mathematics, Wayne State University, Detroit, MI 48202, USA (email: [email protected]). Research of this author was partially supported by the US National Science Foundation under grants DMS-0603846 and DMS-1007132 and by the Australian Research Council under grant DP-12092508. Department of Mathematics, The University of Texas-Pan American, Edinburg, TX 78539-2999, USA (email: [email protected]). 3 Department of Mathematics, The University of Texas-Pan American, Edinburg, TX 78539-2999, USA (email: jsalinasn@broncs. utpa.edu).