A Universal Bound on the Variations of Bounded Convex Functions

作者：Kwon Joon^*

来源：Journal of Convex Analysis, 2017, 24(1): 67-73.

• Convex functions
• variations
• Funk metric
• Thompson metric
• Hilbert metric

Given a convex set C in a real vector space E and two points x, y is an element of C, we investigate which are the possible values for the variation f(y) -f(x), where f : C -> [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 is an element of C.

• 出版日期2017