On convex approximate compact sets and the Efimov-Stechkin spaces (Q1596145)

Let \(M\) be some subset of the Banach space \(X\). The sequence \( \{y_n\}_{n=1}^\infty\subset M \) is called minimizing for the element \( x\in X\), if \( \|y_n-x\|\to \rho(x,M):=\inf\{\|x-y\|\: y\in M\} \) as \( n\to \infty\). The set \(M\) is approximate compact, if for every \( x\in X \) any minimizing sequence contains a subsequence convergent to the element from \(M\). The author constructs an example of a convex bounded approximate compact [see \textit{N.~V. Efimov} and \textit{S. B. Stechkin}, Sov. Math., Dokl. 2, 1226--1228 (1961); translation from Dokl. Akad. Nauk SSSR 140, No. 3, 522-524 (1961; Zbl 0103.08101)] but noncompact set in arbitrary reflexive space.