Given a convex set C in a real vector space E and two points x,yβC,
we investivate which are the possible values for the variation f(y)βf(x),
where f:CβΆ[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 xβC