On locally Chebyshev sets (Q6569645)

Let \(X\) be a normed linear space a non-empty set \(M \subset X\) is said to be Chebyshev, if for all \(x \in X\), the metric projection set \(P_M(x)=\{m \in M: d(x,M)=\|x-m\|\}\) is a singleton. The author earlier [\textit{K. S. Shklyaev}, Sb. Math. 211, No. 3, 455--465 (2020; Zbl 1471.41015); translation from Mat. Sb. 211, No. 3, 158--168 (2020)] has introduced the notion of locally Chebyshev, by requiring for each \(x \in M\) there is a Chebyshev set \(M(x)\) and \(r(x)>0\) such that \(M \cap B(x,r(x)) \subset M(x) \subset M\). In the same work it was shown that connected compact locally Chebyshev set \(M\) in a finite dimensional space, is Chebyshev. In this interesting paper, the infinite dimensional version is established under the additional assumption, \(M\) is boundedly compact, i.e., \(M\) intersects every closed ball in a compact set.