A Universal Bound on the Variations of Bounded Convex Functions

Given a convex set

in a real vector space

E E

and two points

x,y\in C x , y ∈ C

, we investigate which are the possible values for the variation

f(y)-f(x) f ( y ) − f ( x )

, where

f:C\longrightarrow [m,M] 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\in C x ∈ C

.