Applications of Variational Analysis to a Generalized Heron Problem

TL;DR

This paper applies advanced variational analysis techniques to solve a generalized Heron problem, involving finding a point minimizing the sum of distances to multiple sets in a Banach space, with both qualitative and numerical insights.

Contribution

It introduces a variational analysis framework for solving generalized geometric location problems in Banach spaces, extending classical Heron problem solutions.

Findings

Complete solutions in certain Banach space settings

Development of qualitative and numerical methods

Extension of classical Heron problem to multiple sets

Abstract

This paper is a continuation of our ongoing efforts to solve a number of geometric problems and their extensions by using advanced tools of variational analysis and generalized differentiation. Here we propose and study, from both qualitative and numerical viewpoints, the following optimal location problem as well as its further extensions: on a given nonempty subset of a Banach space, find a point such that the sum of the distances from it to $n$ given nonempty subsets of this space is minimal. This is a generalized version of the classical Heron problem: on a given straight line, find a point C such that the sum of the distances from C to the given points A and B is minimal. We show that the advanced variational techniques allow us to completely solve optimal location problems of this type in some important settings.