I am curious about the following question, originated in little discussion I had with a colleague this afternoon.
Let $X,Y$ be complete metric spaces and consider a map $F: X \times Y \to Y$ such that
(i) $F(\cdot, y): X \to Y$ is convex for all $y \in Y.$
(ii) $F(x, \cdot): Y \to Y$ is a contraction with contractivity constant uniform in $x$, i.e., there exists $\rho < 1$ independent of $x$ such that $$d_Y(F(x,y_1),F(x,y_2)) \leqslant \rho(d_Y(y_1,y_2))$$ for all $y_1, y_2 \in Y.$
Now let $y^\ast: X \to Y$ be the function that maps $x \in X$ to the unique fixed point (from Banach Contraction Principle) $y^\ast(x)$ of $F(x, \cdot).$
Question: is $y^\ast$ convex?