Let $H$ be a Hilbert space.

How can one describe continuous mappings $F:H \to H$ that satisfy the following condition:

There exist two elements $c$, $F(c) \neq c$ and a convex compact $M$ containing $c$ and $F(c)$ such that $F(M)$ is convex.

Of course, $F(M)$ is also compact since $F$ is continuous.

Thank you very much in advance!