Let $X$ be a compact Polish space and $Y$ be a separable real Banach space. Assume $U \subseteq X \times Y$ is open, bounded in $Y$-norm, and s.t. for any $x \in X$, $\{y \in Y \mid (x,y) \in U\}$ is convex and nonempty. Does it follow that there is a continuous function $f: X \rightarrow Y$ s.t. the graph of $f$ is contained in $U$?



Every open convex-valued multimap has global sections?
Yes, and this follows directly from a selection theorem in



> 
> Michael, Ernest. “Continuous Selections. I.” Annals of Mathematics,
>  vol. 63, no. 2, 1956, pp. 361–382. Second Series,
>  [www.jstor.org/stable/1969615](http://www.jstor.org/stable/1969615).
> 
> 
> 


Theorem 3.1''' on page 368 shows that a lower hemicontinuous nonempty-valued correspondence from a perfectly normal space to separable Banach space such that all values are convex and, moreover, finite-dimensional, closed, or have an interior point, then the correspondence admits a global section. The statement in the paper contains a typo, as is clear from the statement right after it (the space $\mathcal{K}(Y)$ has to be replaced by $\mathcal{D}(Y)$).


