Mean value theorem for convex functions Let $f$ be a real function with left and right derivatives $f'_-$ and $f'_+$ on the open interval $(a,b)$, and continuous on $[a,b]$ (e.g., let $f$ be convex on $[a,b]$). Then,
Is there something like the mean value theorem for $f$?

Not quite an answer:
For locally Lipschitz functions $f$ there is a notion of generalised gradient $\partial f$ that you can view as a generalisation of the
subdifferential in convex analysis.  The (generalised) mean value theorem then states that there is some $ t \in (a,b)$ such that
${f(b)-f(a) \over b-a } \in \partial f (a + t(b-a))$.
This is not as strong as the corresponding differentiable result.
As an example, take $f(x) = |x|$, $a=-1, b=1$, then
$\partial f(x) = \begin{cases} \{-1\}, & x <0 \\
[-1,1], & x = 0 \\
\{1\}, & x > 0
\end{cases}$,
so we see that
${|1|-|-1| \over 1 - (-1) } = 0 \in \partial f(0) = [-1,1]$.
