Proximinality and diametrically maximal sets in \(C(K)\) (Q627312)

Let \(C\) be a set with diameter \(d\) in a Banach space \(B\). Let \(S[x,r]\) be the closed sphere in \(B\) that is centered at \(x\) and has radius \(r\). Then \(C\) is diametrically maximal if \(C\) is the intersection of the spheres \(S[x,d]\) taken over all \(x\) in \(C\). Let \(DM\) represent the collection of all diametrically maximal sets, and \(DM(d)\) be the sets in \(DM\) having radius \(d\). Let \(E\) be an intersection of closed spheres in \(B\). If \(B\) is the space of continuous real-valued functions on a compact set, this work shows that there is member of \(DM\) that minimizes the Hausdorff metric distance to \(E\). There is also an optimal solution from \(DM(d)\), and, in this case, the authors characterize when the solution is unique.