On proximality pairs (Q2869569)

A pair of subsets \((A,B)\) of a metric space is called a distance pair if there exist \(a\in A\) and \(b\in B\) such that \(d(a,b)=\inf\{ d(x,y): x\in A, y\in B\}\). If such points \(a\) and \(b\) are, in addition, unique then the pair \((A,B)\) is said to be a Chebyshev pair. The paper under review presents a short survey on seminal results about when a given pair of subsets of a metric space is a distance pair and show, as main fact, that whenever \(A\) is a closed convex and locally compact subset of a strictly convex linear space, and \(B\) is a compact subset of that same linear space, then \((A,B)\) is a distance pair.