Convex sets with the Lipschitz fixed point property are compact
Pei-Kee Lin, Yaki Sternfeld · Proceedings of the American Mathematical Society · 1985
Let K K be a noncompact convex subset of a normed space X X . It is shown that if K K is not totally-bounded then there exists a Lipschitz self map f f of K K with inf { ‖ x − f ( x ) ‖ : x ∈ K } > 0 \operatorname {inf}\left \{ {\left \| {x - f\left ( x \right )} \right \|:x \in K} \right \} > 0 , while if K K is totally-bounded then such a map does not exist, but still K 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.