Convex sets with the lipschitz fixed point property are compact
Let K be a noncompact convex subset of a normed space X. It is shown that if K is not totally-bounded then there exists a Lipschitz self map f of K with inf||x - f(x)||: x ∈ K} > 0, while if K is totally-bounded then such a map does not exist, but still K lacks the fixed point property for Lipschitz mappings. It follows that a closed convex set in a normed space has the fixed point property for Lipschitz maps if and only if it is compact. © 1985 American Mathematical Society.
Proceedings of the American Mathematical Society
Lin, P., & Sternfeld, Y. (1985). Convex sets with the lipschitz fixed point property are compact. Proceedings of the American Mathematical Society, 93 (4), 633-639. https://doi.org/10.1090/S0002-9939-1985-0776193-5