A theorem about affine maps in Banach spaces (Q1814110)

The following theorem is proved: If \(C\) is a weakly compact convex set in a strictly convex Banach space, and \(F\) a continuous affine operator on \(X\) having a continuous inverse \(F^{-1}(x)=T^{-1}(x)-T^{-1}(a)\) with \(\| T^{-1}\|\leq 1\), then \(A\cap F(A)\neq\emptyset\) for any subset \(A\) of \(C\) for which \(A\cup F(A)=C\). The theorem generalizes previous results of a similar nature, all of which derive from classical problems such as the Banach-Tarski paradox on decompositon of sets.