Suppose I have two convex functions $f$ and $g$ mapping $\mathbb{R}^{n}\to\mathbb{R}$ (so they are 'more' than proper). Suppose $\|f-g\|_{\infty}<\epsilon$, the sup-norm. In particular, the subdifferentials are bounded, non-empty sets (Theorem 23.4 in Rockafellar). For a given $x$, can we say anything about the Hausdorff distance between $\partial f(x)$ and $\partial g(x)$?
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Maybe if the question isn't answered directly, maybe directing me to a resource that has this type of information would be helpful too. Thanks! – Pallen Oct 16 '14 at 17:04
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I suppose that it is not really reasonable to assume that this is true, given that uniform convergence of differentiable functions does not imply that their derivatives converge. on page 63: https://www.math.ucdavis.edu/~hunter/m125a/intro_analysis_ch5.pdf – Pallen Oct 16 '14 at 17:28
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So this convergence result is true for differentiable convex functions: http://math.stackexchange.com/questions/265930/limit-of-derivatives-of-convex-functions although I'm not sure whether the stronger result of having some distance between the points can be found – Pallen Oct 16 '14 at 17:50
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Apparently it is a theorem named Attouch's Theorem proving the convergence of the subdifferentials given some convergence in the functions called epi-convergence.
On the convergence of subdifferentials of convex functions Hedy Attouch, Gerald Beer http://link.springer.com/article/10.1007%2FBF01207197?LI=true
On the convergence of subdifferentials of convex functions Jean-Paul Penot
http://www.sciencedirect.com/science/article/pii/0362546X9390040Y
Don't know if there are results for quantitative convergence though.
Pallen
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