It is a well-known result that if a sequence of convex function $f_n(\cdot)$ converges on a dense set $C'$ of an open set $C$, then the limit function $f$ exists on $C$, and the converge is uniform over any compacta within $C$. I am concerned with the uniform convergence around the boundary. In $1$-dimension, the question is a classical analysis:
https://math.stackexchange.com/questions/126142/uniform-convergence-of-sequence-of-convex-functions
However I don't find any results in higher dimension, i.e. if a sequence of converx function $f_n(\cdot)$ converges to another continuous convex function $f$ pointwise on a compact convex set $D$, can we obtain uniform convergence over the whole region $D$? (Or, under what conditions on $D$ does the uniform convergence over the whole region is valid?)
@Pietro Majer: I cannot comment due to my current low reputation…What I am looking at is uniform convergence over the whole compact convex $D$ instead of any compacta within the interior. If we are interested in the latter, then Rockafellar has already established the theory. The interesting part is trying to understand boundary convergence property given that the function $f$ is continuous on $D$. Ideally, I don't want a uniform Lipshitz estimate of the sequence since it is often not dorable in practice….
Best Answer
If you have a bound on the uniform norm, say $\|f\|_{\infty, C}\le M$, the sequence has a uniform Lipschitz estimate $ 2M/r$ on the set $C_r \subset C$ of all points with distance at least $r$ from $\partial C$, so by Ascoli-Arzelà a subsequence does converge uniformly on compact sets of the open sets $C$.
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In general, a uniformly bounded sequence $f_n$ of continuous convex functions on a compact convex set $D\subset\mathbb{R}^2$, converging point-wise to a continuous function, need not converge uniformly on $D$. Consider, on the closed unit disk $D$
$$f_n(x,y):=2n^2\Big(x+\frac{y}{n}-1\Big)_+$$
The convex function $f_n$ vanishes on the whole $D$ but a small circular segment $S_n:=\{f_n>0\}$, cut off by the straight line $x+\frac{y}{n}=1$. Note that $\cap_n S_n=\emptyset$. So for any $u\in D$, we have $f_n(u)=0$ eventually. But this convergence to zero is not uniform, because e.g. on the medium point $(x_n,y_n)$ of the arc that bounds $S_n$, $$x_n:=\frac{n}{\sqrt{n^2+1}},\quad y_n:=\frac{1}{\sqrt{n^2+1}}$$ we have $$f_n(x_n,y_n):=2n^2\bigg(\sqrt{1+\frac{1}{n^2}}-1\bigg)=1+o(1)\, .$$