A universal bound on the variations of bounded convex functions

zbMATH Open1360.26009arXiv1401.2104MaRDI QIDQ2974983

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Publication date: 11 April 2017

Abstract: Given a convex set $C$ in a real vector space $E$ and two points $x,yinC$, we investivate which are the possible values for the variation $f(y)-f(x)$, where $f:Clongrightarrow[m,M]$ is a bounded convex function. We then rewrite the bounds in terms of the Funk weak metric, which will imply that a bounded convex function is Lipschitz-continuous with respect to the Thompson and Hilbert metrics. The bounds are also proved to be optimal. We also exhibit the maximal subdifferential of a bounded convex function at a given point $xinC$.

Full work available at URL: https://arxiv.org/abs/1401.2104

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