Reference request: concave/convex envelope I'm seeking the references concerning on the regularity analysis of concave envelopes, i.e. given some measurable function $f:\mathbb R^d\to\mathbb R$ that is bounded from above ($d=1$ or $d\ge 1$), denote by $f_{\#}$ its concave envelope, i.e.


*

*$f_{\#}:\mathbb R^d\to\mathbb R$ is concave and $f_{\#}\ge f$;

*For any concave function $g:\mathbb R^d\to\mathbb R$ s.t. $g\ge f$, one has $g\ge f_{\#}$.


Assume that $d=1$. I'm mainly interested in the boundedness of the (left/right) derivative $f_{\#}'$. Under what kind of conditions on $f$, we can show that $f_{\#}'$ is uniformly bounded? Any related reference is welcome! Thanks a  lot!
 A: Thanks Paata Ivanisvili, BD. and Dirk for their reply and I'd like to summarize what I'm thinking about this issue. I claim that it is not an answer. 
Assume that $f$ is bounded from above and $L-$Lipschitz. To simplify assume further $d=1$ and $f_{\#}$ is differentiable. Then one has $\|f_{\#}'\|\le L$. If not, assume that there exists $x_0$ s.t. $f_{\#}'(x_0)=M>L$, then $f_{\#}'(x)\ge M>L$ for all $x\le x_0$, and thus
$$\limsup_{x\to-\infty}\big(f_{\#}(x)-f(x)\big)~~=~~-\infty.$$ 
If otherwise there exists $x_0$ s.t. $f_{\#}'(x_0)<-L$, one has as well 
$$\limsup_{x\to+\infty}\big(f_{\#}(x)-f(x)\big)~~=~~-\infty.$$
These two situations contradict the definition of $f_{\#}$. 
I look forward to the further analysis on the regularity of $f_{\#}$, once $f$ is supposed bo be very "nice". Finally, I find the following paper which may be useful for who is also interested in this issue:
https://people.math.ethz.ch/~afigalli/papers/Optimal%20regularity%20of%20the%20convex%20envelope.pdf
A: I think the Benson's paper can be a good reference.
You can find it here:
http://link.springer.com/article/10.1023/B:COAP.0000004976.52180.7f
