The best approximation pair problem relative to two subsets in a normed space

TL;DR

This paper extends the best approximation pair problem to general normed spaces and non-convex sets, providing new conditions for existence and uniqueness, and broadening its scope across scientific fields.

Contribution

It introduces geometric conditions for the existence and uniqueness of BAP solutions in general normed spaces with non-convex sets, expanding previous results.

Findings

Several sufficient geometric conditions for BAP uniqueness.

Many conditions for the existence of a BAP.

Extension of algorithms to broader settings.

Abstract

In the classical best approximation pair (BAP) problem, one is given two nonempty, closed, convex and disjoint subsets in a finite- or an infinite-dimensional Hilbert space, and the goal is to find a pair of points, each from each subset, which realizes the distance between the subsets. We discuss the problem in more general normed spaces and with possibly non-convex subsets, and focus our attention on the issues of uniqueness and existence of the solution to the problem. As far as we know, these fundamental issues have not received much attention. We present several sufficient geometric conditions for the (at most) uniqueness of a BAP. These conditions are related to the structure and the relative orientation of the boundaries of the subsets and to the norm. We also present many sufficient conditions for the existence of a BAP. Our results significantly extend the horizon of a recent algorithm for solving the BAP problem [Censor, Mansour, Reem, J. Approx. Theory (2024)]. The paper also shows, perhaps for the first time, how wide is the scope of the BAP problem in terms of the scientific communities which are involved in it (frequently independently) and in terms of its applications.