directional derivatives and convex sets

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# directional derivatives and convex sets

Submitted by sjar on Tue, 11/13/2012 - 16:50

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Assume U \subset R^n is convex (not necessarily open) and all partial derivatives (first order) exist in all points of U. Furthermore all partial derivatives are bounded by the same number M >= 0. Prove that for all x, y \in U:

|f(x) - f(y)| <= M \sqrt{k} ||x - y||_2

Note:

It is rather easy, when U is multi-dimensional interval (one can estimate |f(x) - f(y)| coordinate-wise), but I can't come up with generalization.

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