Let $f:X\times Y\to \mathbb{R}$ be a bounded continuous function on the product topological space $X\times Y$. It is well known that if $Y$ is compact, then the infimum function $g(x) = \inf_{y\in Y}f(x,y)$ is continuous on $X$. Now I let $\mathcal{X}$ be a convex compact subset of a finite dimensional Hilbert space and let $\mathcal{Y}$ be a convex closed and bounded subset of a infinite dimensional Hilbert space. The functional $F:\mathcal{X}\times \mathcal{Y}$ is continuous and strongly convex on $\mathcal{Y}$ for every element $x\in\mathcal{X}$. Then the function $$G(x) = \inf_{y\in\mathcal{Y}} F(x,y)$$ is well-defined (see Unique minimizer of a quadratic functional $f(x)= \frac{1}{2}||x||^2-\phi(y)$).
Is $G(x)$ a continuous function? If not, is there a weaker enhacement on $\mathcal{Y}$ other than compactness to guarantee continuity?