Relative Chebyshev Centers in Hilbert Space
چکیده
We characterize inner product spaces in terms of the relationship between the relative Chebyshev center of a set and the absolute Cheby-shev center of an associated set. A consequence of the characterization is that an existing nite algorithm can be adapted to calculate the center of a nite set relative to a nite dimensional aane space. We also discuss the case where the constraint set is nonlinear. If a subset of a normed linear space has a smallest possible bounding sphere, the center of such a sphere is called a Chebyshev center for the set. One application of the theory of Chebyshev centers is to the problem of nding the location of a central communications facility to minimize the maximum response time to a set of remote locations. The original motivation for the present paper was the desire to compute the best center in a convex constraint set. Suppose (X; kk) is a real Banach space and K X is closed and convex. If A is a bounded subset of X, let Z K (A) := fw 2 K : sup a2A kw ? ak = inf u2K sup a2A ku ? akg We call Z K (A) the K-relative Chebyshev center of A. The X-relative Cheby-shev center of A is called the (absolute) Chebyshev center of A and is denoted by Z(A).