Convergence of the infima of convex functions on $\mathbb{R}^m$

Question

Any thoughts on proving the following statement, which is a generalization of the result in convergence of the infima of convex functions from domain $\mathbb{R}$ to $\mathbb{R}^m$ and also Theorem 1 from Pinelis - A Necessary and Sufficient Condition on the Stability of the Infimum of Convex Functions?

Suppose $f_n: \mathbb{R}^m \rightarrow (-\infty, \infty]$ is a sequence of convex functions that converges pointwise to a convex function $f: \mathbb{R}^m \rightarrow (-\infty, \infty]$ that is not monotonic in every coordinate of $\mathbb{R}^m$, then $\inf_{x \in\mathbb{R}^m} f_n(x)$ converges to $\inf_{x\in\mathbb{R}^m} f(x)$.

Answer 1

1. $\newcommand\R{\mathbb R}$It does not make sense to define a convex function on the nonconvex set $[-\infty,\infty]^m$.

2. Even if you replace $[-\infty,\infty]^m$ by $\mathbb R^m$ and even you just take $m=1$, then your desired statement will still be false in general. For instance, do take $m=1$ and let $f(x):=e^x$ and $f_n(x):=e^x\,1(x\ge-n)+e^{-n}(x+n+1)\,1(x<-n)$ for real $x$ and natural $n$. Then $f$ and the $f_n$'s are convex, $f_n\to f$ pointwise on $\R$ (as $n\to\infty$), but $$\inf_\R f_n=-\infty\not\to0=\inf_\R f.$$

A necessary and sufficient condition for the infimum-stability for convex functions $f$ defined on $\R$ was given by Theorem 1 of A Necessary and Sufficient Condition on the Stability of the Infimum of Convex Functions (one can also use the arXiv version).

The proof of the just mentioned result takes about 4 pages. No extension of that result to $m>1$ seems to be known.

Answer 2

There is a general theory that deals with exactly this problem: When do minimizers of sequences of functions converge and if they converge, when is the limit again a minimizer (and of what)? The important notion here is $\Gamma$-convergence. I can recommend the book "A. Braides: Γ-convergence for beginners. Oxford University Press, 2002."

In a nutshell: If a sequence $f_n$ and another function $f$ fulfill

1. For every convergent sequence $x_n\to x$ it holds that $$f(x)\leq \liminf f_n(x_n),$$ and
2. for every $x$ there exists a sequence $x_n$ with $x_n\to x$ and $$f(x)\geq \limsup f_n(x_n),$$

then it holds that every cluster point of a sequence $x_n$ of minimizers of $f_n$ will be a minimizer of $f$. Convexity does not play a role here, btw.