Bounded nonconvex Chebyshev sets in a real inner product space.
by
Gordon G. Johnson
University of Houston

A subset S, of a real finite dimensional inner product space X, has the property that each point in X has a unique nearest point in S iff S is closed and convex. Such a set is called a unique nearest point set.

If X is a Hilbert space, must a unique nearest point set in X be convex? The real inner product space Y of all infinite number sequences having at most a finite number of nonzero terms, with the usual inner product, contains a non convex unique nearest point set S, moreover S can be made bounded.

Now, if H denotes the completion of Y, and T the closure of a bounded unique nearest point set S in Y, then if each of {Pi} and {Qi} is a sequence in Y, converging to points P and Q respectfully in H, where for each i, Qi is the unique nearest point in S to Pi, then Q is the unique nearest point to P in T. Moreover, there is a sequence {Vi} in Y converging to a point V in H and a non convergent sequence {Wi}, where Wi is the unique nearest point to Vi in S.

Date received: August 25, 2002