An invitation to the study of a uniqueness problem

TL;DR

This paper motivates the study of a uniqueness problem in real normed spaces, focusing on conditions ensuring the uniqueness of the global minimum of a specific function related to a set and a mapping.

Contribution

It introduces a new problem in the context of normed spaces, proposing conditions for the uniqueness of minima of a particular function involving a set and a mapping.

Findings

Proposes a new uniqueness problem in normed spaces.

Highlights the importance of conditions for the function's global minimum.

Sets the stage for future research on the problem.

Abstract

In this very short paper, we provide a strong motivation for the study of the following problem: given a real normed space $E$, a closed, convex, unbounded set $X\subseteq E$ and a function $f:X\to X$, find suitable conditions under which, for each $y\in X$, the function $$x\to \|x-f(x)\|-\|y-f(x)\|$$ has at most one global minimum in $X$.