Nearest points and delta convex functions in Banach spaces

TL;DR

This paper surveys the properties of nearest points in Banach spaces and explores the relationship between delta-convex functions and the problem of locating nearest points in closed sets.

Contribution

It provides a concise overview of the set of points with nearest points in Banach spaces and examines the connection to delta-convex functions.

Findings

Characterization of points with nearest points in Banach spaces

Relationship between delta-convex functions and nearest point problems

Insights into the size of the set of points with nearest points

Abstract

Given a closed set $C$ in a Banach space $(X, \|\cdot\|)$, a point $x\in X$ is said to have a nearest point in $C$ if there exists $z\in C$ such that $d_C(x) =\|x-z\|$, where $d_C$ is the distance of $x$ from $C$. We shortly survey the problem of studying how large is the set of points in $X$ which have nearest points in $C$. We then discuss the topic of delta-convex functions and how it is related to finding nearest points.