Can one give a reference to a result like this:
- If a sequence of convex functions $f_{n}$ on $\mathbb{R}$ converges pointwise to a non-monotonic function $f$, then $\displaystyle\inf_{\mathbb{R}} f_n$ converges to $\displaystyle\inf_{\mathbb{R}} f$?
Thank you.
I don't know a reference to a result. But the statement is simple enough to just give a proof here.
First observe that $f$ is necessarily convex (taking the limit of $f_i(tx + (1-t)y) \leq tf_i(x) + (1-t)f_i(y)$), and non-monotonic. This implies that $f$ attains its infimum [Roc, Section 27]: there exists $x_0$ such that $f(x_0) \leq f(x)$ for all $x\in \mathbb{R}$.
We now prove the claim by contradiction. Assume that $\inf f_i \not\to \inf f$. Then at least one of $\limsup_i \inf_{\mathbb{R}} f_i > \inf f$ or $\liminf_i \inf_{\mathbb{R}} f_i < \inf f$ is true, we call them cases 1 and 2.
Case 1: after taking a subsequence $f_{i'}$ we have that there exists a fixed constant $\epsilon$ such that $f_{i'}(x) > \inf_{\mathbb{R}} f_{i'} > \inf f + \epsilon$. This contradicts $f_{i'}(x_0) \to f(x_0) = \inf f$.
Case 2: after taking a subsequence $f_{i'}$ we have that there exists a fixed constant $\epsilon$ such that $\inf_{\mathbb{R}} f_{i'} < \inf f - \epsilon$. By definition there exists $x_{i'}$ such that $f_{i'}(x_{i'}) \leq \inf f_{i'} + \epsilon / 2$. Now, if $x_{i'}$ accumulates at $x_{\infty}$, then $f(x_{\infty}) \leq \inf f_{i'} + \epsilon / 2 < \inf f$ is a contradiction. So $x_{i'}$ cannot accumulate, so that only finitely many can belong to every closed interval. Without loss of generality we can assume that a subsequence $x_{i''} \to +\infty$. By convexity $f_{i''}|_{[x_0,x_{i''}]} \leq \inf f$ hence pointwise convergence implies that $f |_{[x_0,\infty)} = f(x_0)$. This contradicts non-monotonicity of $f$.
[Roc]: Rockafeller, Convex Analysis
Note that I've assumed that non-monotonicity means non-weak-monotonicity. If non-monotonicity allows functions that are weakly, but not strongly, monotonic, then the statement is false: if you require $\inf f_n > -\infty$, you can let $f(x) = 0$ if $x < 0$ and $f(x) = x$ for $x\geq 0$. And define $f_n$ by
$$ f_n(x) = \begin{cases} -1 & x < -n \\ x / n & -n \leq x < 0 \\ x & 0 \leq x \end{cases} $$
I have since found a few results in the literature, which are somewhat similar to what I had in mind, but not quite the same.
Of course, if the functions $f$ and $f_n$ are assumed to be real-valued, then various simple proofs of the result that I stated can be given; in particular, then the convex functions $f$ and $f_n$ are necessarily continuous, and so, provided that the functions are not monotonic, their infima are attained and hence can be replaced by the corresponding minima. However, in the applications that I had in mind, it is necessary to allow the functions $f$ and $f_n$ to take the value $\infty$ as well, and then the proof becomes significantly more complicated.
Details on the above statements can be found in the note ``A necessary and sufficient condition on the stability of the infimum of convex functions'' that I have just posted at http://arxiv.org/pdf/1307.3806v3.pdf .
