Lim's center and fixed-point theorems for isometry mappings

In this article, we prove that if K is a nonempty weakly compact convex set in a Banach space such that K has the hereditary fixed-point property (FPP) and ℱ is a commuting family of isometry mappings on K, then there exists a point in C(K) which is fixed by every member in ℱ whenever C(K) is a compact set. Also, we give an example to show that C(K), the Chebyshev center of K, need not be invariant under isometry maps. This example answers the question as to whether the Chebyshev center is invariant under isometry maps. Furthermore, we give a simple example to illustrate that Lim's center, as introduced by Lim, is different from the Chebyshev center.