Let $X$ be a locally convex TVS, and let $A$ and $B$ be convex and compact subsets of $X$ with $A \subset B$.
Let $f: A \times B \to [0,\infty]$ have the following properties:
(1) For all $b \in B$, $f(\cdot, b)$ is lower semicontinuous.
(2) For all $b \in B$, $f(\cdot, b)$ is convex.
(3) For all $a \in A$, $f(a, \cdot)$ is continuous.
(4) For all $a \in A$, the function $f(a, \cdot)$ is minimized uniquely at $a$, i.e. $f(a,a) < f(a,b)$ for all $b \neq a$.
Does it follow that for all $b \in B$, the function $g_b: A \to [0,\infty]$ defined by $g_b(a) = f(a,b) - f(a,a)$ attains a minimum?
Note that by (4), $g_b$ is bounded below by $0$, so $\inf g_b(A)$ is finite for all $b \in B$. If $b \in A$, then the result is trivial, for then $g_b$ is minimized (uniquely) at $b$, again by (4). In general, it would be sufficient to show that $g_b$ is lower semicontinuous, but I have not been able to see that that's the case.