A counterexample on global Chebyshev approximation (Q1106416)

Let Y be a non-empty subset of a B-space X and \(x\in X\). Then \(y\in Y\) is called a best approximation to x if \(\| x-y\| =dist(x,Y)\). Y is said to be proximinal if each \(x\in X\) has at least one best approximation in Y. If A,Y are subsets of a B-space X then the Chebyshev radius and the Chebyshev centre of A relative to Y are defined respectively as follows: \(r_ Y(A)=\inf_{y\in Y}\sup_{a\in A}\| a-y\|\) and \(E_ Y(A)=\{y\in Y:\sup_{a\in A}\| a-y\| =r_ Y(A)\}\). If \(Y=X\) we simply speak about the Chebyshev radius r(A) and the Chebyshev centre E(A) of A, respectively. It has been shown by Franchetti and Cheney that the Chebyshev centre of any bounded subset of C(S) (where S is a compact Hausdorff space) is non- empty. This is equivalent to answering whether or not Y is proximinal in C(S\(\times T)\) for each compact Hausdorff space T. In this paper the author constructs a counterexample to answer this question in the negative.