A note on sets with the unique farthest point property (Q1062225)

The existence of nonsingleton sets with the unique farthest property is equivalent, in infinite-dimensional Hilbert spaces, to the (possible) existence of nonconvex Chebyshev sets, so it is not known whether a similar set can exist. The main result of this paper is the following: if an infinite dimensional Hilbert space H contains a nonsingleton, uniquely remote subset G, then it necessarily contains also sets, say G', for which diam(G') is arbitrarily close to twice the Chebyshev radius. (Reviewer's remark: Recently, Gordon G. Johnson claimed to have constructed an example of a nonconvex Chebyshev set in an incomplete inner product space.)