Examples of sets with given approximation properties in \(WCG\)-space (Q2439972)

A nonempty subset \(M\) of a Banach space \(X\) is called approximatively compact if for every \(x\in X\) every sequence \((y_n)\) in \(M\) such that \(\|y_n-x\|\to \rho(x,M):=\inf\{\|x-y\| : y\in Y\}\) contains a subsequence converging to some \(z\in M\). It is obvious that this element is nearest to \(x\) in \(M,\) written as \, \(z\in P_M(x).\) The approximative compactness is a very important notion in best approximation theory -- existence and uniqueness, continuity properties of the metric projection \(P_M\). A question studied by several authors is the existence of approximatively compact, not compact sets. In this paper, the author shows that every infinite-dimensional weakly compactly generated (WCG) Banach space contains an approximatively compact not locally compact set. In the case of separable infinite-dimensional Banach spaces (which are WCG), such examples were found by the author [Russ. Math. Surv. 49, No. 4, 153--154 (1994); translation from Usp. Mat. Nauk 49, No. 4(298), 157--158 (1994; Zbl 0883.46008)] (approximatively compact not compact) and \textit{I. A. Pyatyshev} [Russ. Math. Surv. 62, No. 5, 1007--1008 (2007); translation from Usp. Mat. Nauk 62, No. 5, 163--164 (2007; Zbl 1147.46019)] (approximatively compact not locally compact). The author [Mosc. Univ. Math. Bull. 54, No. 4, 18--20 (1999); translation from Vestn. Mosk. Univ., Ser. I 1999, No. 5, 19--21 (1999; Zbl 0974.41011)] has also shown that every reflexive Banach space (which is WCG, too) contains an approximatively compact convex body. Two related open problems are posed: the existence of non-trivial Chebyshev sets in any Banach space, and the existence of an approximatively compact not boundedly compact set in any infinite-dimensional Banach space.