Fixed points of isometries on weakly compact convex sets (Q1399367)

Let \(K\) be a nonempty convex set in a Banach space and \(T:K\to K\) an isometry. If \(K\) is closed and bounded and if \(X\) is uniformly convex, then it is shown that the Chebyshev-center is a fixed point of \(T\). If \(X\) is arbitrary and \(K\) is weakly compact such that each nonempty closed convex subset has the fixed point property for isometries, then \(T\) has a fixed point in the Chebyshev-center of \(K\).