Extending and paralleling Stechkin's category theorem (Q2075667)

If \(K\) is a compact subset of a metric space \((X,d)\) and \(x\in X\setminus K\) then the function \(y\mapsto d(y,x)\) takes on a minimum and maximum value on~\(K\). So the sets \(p_K(x)\) and \(F_K(x)\), of points where that minimum and maximum, respectively, are attained are nonempty.\par The Category Theorem of Stechkin mentioned in the title says that in Euclidean spaces the set~\(p_K(x)\) is a singleton for residually many~\(x\) [\textit{S. B. Stechkin}, Acad. Republ. Popul. Roum., Rev. Math. Pur. Appl. 8, 5--18 (1963; Zbl 0198.16202)]. The author surveys many generalizations of this result and provides new versions for length spaces. A length space is a metric space~\((X,d)\) where for each pair of points~\(a\) and~\(b\) there is a segment of length \(d(a,b)\) that joins them. The author proves that in a complete length space that satisfies an additional technical condition the nearest point map is single-valued for a residual set of points, and for a residual set of compact sets the farthest point map is multi-valued on a dense set and single-valued on a residual set.