# Does this function attain a minimum?

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.

• 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... – Zim Sep 9 at 15:51
• 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! – aduh Sep 15 at 3:46
• 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? – Zim Sep 15 at 19:51
• @ 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? – aduh Sep 15 at 20:57
• Hmmm, oh yeah that is a good point! I think I was mistaken, sorry :( – Zim Sep 16 at 22:50