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.

  • $\begingroup$ You might have luck using results on Marginal functions here... in particular, lower semicontinuity of $f\colon X\times Y\to\mathbb{R}$ is inherited by the marginal function $F:x \mapsto \inf f(x, Y)$ [Lemma 1.30, Bauschke & Combettes' book, vol. 2]. However, it's not clear to me that $f$ is necessarily lsc to begin with... $\endgroup$ – Zim Sep 9 at 15:51
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    $\begingroup$ Thanks, I explored this a bit, but, so far as I can tell, even if I knew that $f$ were lsc in the product topology, this would only allow me to conclude that $a \mapsto f(a,a)$ is lsc, whereas I actually need it to be usc. Please do let me know if you had something else in mind! $\endgroup$ – aduh Sep 15 at 3:46
  • $\begingroup$ I wonder if continuity is preserved via margnal function? If so, then that with (3) should show that $a\mapsto f(a,a)$ is continuous. Then, since $a\mapsto f(a,b)$ is lsc I think that would get it. If continuity is not preserved in general, then perhaps it might be preserved by this particular function? $\endgroup$ – Zim Sep 15 at 19:51
  • $\begingroup$ @ Zim I'm not sure I follow the suggestion. Continuity preservation in the sense of Lemma 1.30 would require continuity of $f$ in the product topology. Why would (3) be enough to infer that $a \mapsto f(a,a,)$ is continuous? $\endgroup$ – aduh Sep 15 at 20:57
  • $\begingroup$ Hmmm, oh yeah that is a good point! I think I was mistaken, sorry :( $\endgroup$ – Zim Sep 16 at 22:50

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