Let $K$ be a convex body in $ R^d$. (Say, a ball, say a cube...) For which classes $ \cal C$ of functions, every function $ f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$.

### motivation:

Of course this holds for continuous functions so we want more general classes of functions. For $d=1$ we ask for extensions of the intermediate value function, and the first such extension known to me is a theorem of Darboux which gives the intermediate value function for differentials of continuous functions. I asked about extensions for Darboux theorem in this post, and Marton Elekes proved it for high dimensional cubes.I recalled the question again after hearing a lecture by Haim Brezis, where he described three classes of functions that satisfy the intermediate valye theorem for $d=1$. He has other results about when you can "hear" the degree of a (non-contiuous) map which seems very relevant. (See Haim's publication list for some relevant papers with Nirenberg, Bourgain, Mironescu, Nguyen, and others.)

### Warning:

The answer may depend on $K$.