Solving a Generalized Heron Problem by means of Convex Analysis

TL;DR

This paper extends the classical Heron problem to a broader convex analysis framework, solving for a point that minimizes the sum of distances to multiple convex sets in higher-dimensional spaces.

Contribution

It introduces a generalized Heron problem within convex analysis and provides a solution method for minimizing the sum of distances to convex sets.

Findings

Provides a convex analysis-based solution to the generalized Heron problem.

Extends classical geometric problem to higher dimensions and convex sets.

Offers theoretical insights applicable to optimization and computational geometry.

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 $\R^s$, find a point such that the sum of the distances from that point to $n$ given nonempty closed convex subsets of $\R^s$ is minimal.