Solving a Generalized Heron Problem by means of Convex Analysis

Abstract

The classical Heron problem states: \emph{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 ${\mathbb{R}}^s$, find a point such that the sum of the distances from that point to $n$ given nonempty closed convex subsets of ${\mathbb{R}}^s$ is minimal.