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Let $\mathcal{S}$ be the set of proper convex functions functions from $X$ to $\mathbb{R}$, where $X$ is a open and convex subset of $\mathbb{R}^{n}$. I was wondering under which conditions on $\mathcal{S}$ we have \begin{gather} \forall \epsilon>0\ \exists \delta>0 \text{ such that for } f,h \in \mathcal{S} ,\ ||f-h||_{\infty} < \delta \\ \Rightarrow \underset{ x \in X}{\sup} \underset{ v \in \partial f(x), w \in \partial h(x)}{\sup}||v-w||_2 <\epsilon \end{gather}

Motivation: Intuitively, the fact that $||f-h||_{\infty}$ is small, means that the shape of the graphs of the functions are similar and hence also their suporting hyperplanes might be similar. Of course this is just a picture that I have in mind for the 1-dimensional case.

Any help or suggestions for possible references to similar results would be great.

From where the problemm comes: I have a function $F$ that maps convex functions to elements of their subgradient at any given point. I would like to show that if the mapped functions are similar, i.e. $||f-h||_{\infty}$ is sufficiently small, then we can say that $||F(h)-F(f)||_{2}$ is small.

sigmatau
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1 Answers1

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Assuming any conditions on $S$ if $S$ still contains at least one non-smooth function then your claim does not hold.

Proof: Let $f \in S$ such that $f$ is not differentiable at $x_0 \in X.$ Hence there exist $v , w \in \partial f(x_0) $ such that $ v \neq w $. Set $ \epsilon = \| v -w \| $ Then observe that for any positive delta , your statement does not holds for two functions $f,~ h$ where $ f = h $ and $v \in \partial f(x_0) $ and $ w \in \partial h(x_0) $.

P.S: But I feel like your claim would be true if you switch the second $\sup$ to $\inf$.

Red shoes
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  • I found this ( https://math.stackexchange.com/questions/976913/hausdorff-distance-between-subdifferential-sets?rq=1). Hence, there exists a notion of convergence of function which guarantees that the subdifferentials converge in some sense, but I have to give it a look. – sigmatau Nov 18 '18 at 20:45
  • are you confident that the result holds if we switch $\sup$ with $\inf$. Do you have an idea how to prove that? – sigmatau Nov 19 '18 at 08:33
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    @sigmatau I'm not 100 % sure about. I just guessed , that's what my intuition tells me. I need to pick up pen I spend time to prove it (or reject it). I suggest you write a separate question for that. – Red shoes Nov 19 '18 at 17:17
  • Already did it today https://math.stackexchange.com/questions/3004735/a-condition-for-similarity-of-subgradients-of-convex-proper-functions . Thank you for your help. . – sigmatau Nov 19 '18 at 17:22