Geodesic
A Universal Bound on the Variations of Bounded Convex Functions
Journal of convex analysis, Tome 24 (2017) no. 1, pp. 67-73
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Given a convex set $C$ in a real vector space $E$ and two points $x,y\in C$, we investivate which are the possible values for the variation $f(y)-f(x)$, where $f:C\longrightarrow [m,M]$ 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 $x\in C$.
Classification : 26B25, 52A05
Mots-clés : Convex functions, variations, Funk metric, Thompson metric, Hilbert metric 
@article{JCA_2017_24_1_JCA_2017_24_1_a4,
     author = {J. Kwon},
     title = {A {Universal} {Bound} on the {Variations} of {Bounded} {Convex} {Functions}},
     journal = {Journal of convex analysis},
     pages = {67--73},
     year = {2017},
     volume = {24},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/JCA_2017_24_1_JCA_2017_24_1_a4/}
}
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J. Kwon. A Universal Bound on the Variations of Bounded Convex Functions. Journal of convex analysis, Tome 24 (2017) no. 1, pp. 67-73. http://geodesic.mathdoc.fr/item/JCA_2017_24_1_JCA_2017_24_1_a4/