maximum of a multivariate function on the boundary

Question

I have a smooth function of 2 variables: $F(x,y)$ with domain $D$, which is a compact set. If for every fixed $y_0$, $F(x,y_0)$ is convex, then can I claim the maximum of $F(x,y)$ is on the boundary of $D$? not interior point of $D$?

Answer 1

Uniqueness of the maximizer of $F$ over $D$ is necessary. If $(x_*,y_*)$ is the maximizer of $f$ over $D$, then $f(x,y_*)$ is convex with maximizer on the $x$-boundary .Then claim that the maximizer is the same as $x_*$ which shows the desirable result.

If $F$ has multiple maximizer, the results might not be true, e.g., $F(x,y)=-y^2$ and $D=[-1,1]\times [-1,1]$.