Best simultaneous approximations (Chebyshev centers)

The problem of approximating simultaneously a set of data in a given metric space by a single element of an approximating family arises naturally in many practical problems. A common procedure is to choose the ''best'' approximant by a least squares principle, which has the advantages of existence, uniqueness, stability and easy coraputability. However, in many cases the least deviation principle makes more sense. Geometrically, this amounts to covering the given data set by a ball of minimal radius among those centered at points of the approximating family. The theory of best simultaneous approximants in this sense, called also Chebyshev centers, was initiated by A. L. Garkavi about twenty years ago. It has drawn more attention in the last decade, but is still in a developing stage. In this short survey I try to describe the main known results and to point at some of the connections between the theory of Chebyshev centers and other problems of Approximation Theory and of Banach Space Theory.