The center of a convex set

Abstract

Let X be a Banach space and K a weakly compact convex nonvoid subset with normal structure (1]. Brodskii and Mil'man (1] constructed, using transfinite induction, a center of K which is fixed by every isometry mapping K onto K. In this note, we construct a unique center for a weakly compact convex nonvoid subset (not necessarily having normal structure) which is fixed by every affine isometry mapping K into K. A similar theorem for weak* compact convex sets is also possible under some additional assumptions. CONSTRUCTION. Let K be a nonempty weakly compact convex subset of a Banach space. We shall define C. for all ordinals a by transfinite induction. Set CO = K. Let /8 be an ordinal and suppose that C. has been defined for a < / in such a way that (i) each Ca is a nonempty closed convex subset of K and (ii) Ca, a < /3, is decreasing. If /3 is a limit ordinal, we setC = n a<,Ca. Otherwise, let y be the predecessor of /8 and let