A note on ''Approximation of bounded sets''
Let X be a normed linear space, V and F non-empty subsets of X with V convex and F bounded. An element \(v_ 0\in V\) is called best simultaneous approximation (b.s.a.) to F if \(\sup_{x\in F}\| x-v_ 0\| =\inf_{v\in V}\sup_{x\in F}\| x-v\|.\) Let K be a \(\sigma (V,X^*)\)-compact subset of the dual unit ball \(B(X^*)\) which norms F- V i.e. such that \(\| x-v\| =\sup_{f\in K}f(x-v)\) for all \(x\in F\), \(v\in V\). On the compact space (K,\(\sigma\) (V,K)), consider the upper semi-continuous function \[ U^+_ F(f)=\limsup_{K\ni g\to \sigma (V,K)}f\sup_{x\in F}g(x)=\inf \sup_{g\in W}\sup_{x\in F}g(x) \] over all \(\sigma\) (V,K)-neighbourhoods W of f in K. The following characterization of b.s.a. (Chebyshev centre) was given by \textit{J. H. Freilich} and \textit{H. W. McLaughlin} in the paper cited in the title [ibid. 34, 146-158 (1982; Zbl 0504.41013)] - Theorem 2: Let \(v_ 0\in V\) and \(E=\sup_{f\in F}\| f-v_ 0\|.\) Then \(v_ 0\) is a b.s.a. if and only if for each \(v\in V\) there exists an \(L\in ext(K)\) such that (i) \(U^+_ F(L)-L(v_ 0)=E,\) and (ii) \(L(v-v_ 0)\leq 0.\) By showing that the necessity part of this theorem is false, the following correct version is given in the paper under review: Theorem. \(v_ 0\in V\) is a b.s.a. of F in V if and only if for every \(v\in V\) there is an L in the \(\sigma\) (V,K)-closure of ext(K) which satisfies (i) and (ii) above.
scientific article; zbMATH DE number 3973504