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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....

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  • $\begingroup$ Roy, prompted by another user, I requested that two unregistered accounts of yours be merged. This brings you nearer to the reputation you need to comment. But if you plan on becoming part of the community (I hope you do!), please consider registering your account. Of course it is entirely up to you. $\endgroup$
    – Todd Trimble
    Commented Feb 15, 2015 at 11:55
  • $\begingroup$ @Roy Han: I think dangerous points are the accumulation points of extreme points, a case that of course can't occur in 1-dimension. I think the 1-dimensional case generalizes to convex polytopes. $\endgroup$
    – Pietro Majer
    Commented Feb 15, 2015 at 14:35
  • $\begingroup$ @ToddTrimble Thanks for the reminder! I officially registered and it seems that I can comment now even if I still have 45 points of reputation.... $\endgroup$
    – Roy Han
    Commented Feb 15, 2015 at 19:36
  • $\begingroup$ @PietroMajer Can you be more specific about the meaning of 'accumulation points of extreme points'? I am thinking of a fixed, let's say, compact polytope. $\endgroup$
    – Roy Han
    Commented Feb 15, 2015 at 19:37
  • $\begingroup$ I mean any point which is a limit of a non-constant sequence of extreme points of D. E.g. any point of the boundary if D is a closed disk. In this case the example below can be repeated. $\endgroup$
    – Pietro Majer
    Commented Feb 15, 2015 at 19:43

1 Answer 1

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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 $$f_n(x_n,y_n):=2n^2\bigg(\sqrt{1+\frac{1}{n^2}}-1\bigg)=1+o(1)\, .$$

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  • $\begingroup$ I vaguely think that analogous examples of loss of uniform convergence to a continuous limit, can always be shown for any convex $D$ and any $p\in \partial D$ which is an accumulation point of extreme points. $\endgroup$
    – Pietro Majer
    Commented Feb 15, 2015 at 14:11
  • $\begingroup$ Around isolated accumulation points probably one can show uniform convergence, if the limit function is continuous, analogously to the 1 dim case. $\endgroup$
    – Pietro Majer
    Commented Feb 15, 2015 at 14:16
  • $\begingroup$ Isn't $f(x_n,y_n)=1/(n(\sqrt{n^2+1}+n))=O(n^{-2})$? $\endgroup$
    – Roy Han
    Commented Feb 15, 2015 at 19:42
  • $\begingroup$ I think I deleted by mistake a factor $n^2$ in the definition of $f_n$ :) fixed now. $\endgroup$
    – Pietro Majer
    Commented Feb 15, 2015 at 19:53
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    $\begingroup$ That's a very interesting point indeed. It seems as long as some local curvature exists some phenomenon would happen.... $\endgroup$
    – Roy Han
    Commented Feb 15, 2015 at 19:54

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