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.
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 $\inf$.