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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.

  1. $f_{\#}:\mathbb R^d\to\mathbb R$ is concave and $f_{\#}\ge f$;
  2. 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!

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  • $\begingroup$ It depends very much on each particular $f$. On $\mathbb{R}$ we should first require that concave envelope exists: the graph of $f$ should be below a linear function $f(x)\leq Ax+B$. Besides, we should require $f$ is uniformly Lipschitz in order to expect that the envelope is uniformly Lipschitz. I believe these 2 conditions will be also sufficient. $\endgroup$ Nov 7, 2016 at 15:25
  • $\begingroup$ The paper: "The Dirichlet Problem for the Degenerate Monge--Amp`ere equation", L. Caffarelli, L. Nirenberg, J. Spruck, studies regularity properties of the concave envelopes for obstacles $f$ given on the boundary of a convex set. $\endgroup$ Nov 7, 2016 at 15:34

2 Answers 2

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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

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  • $\begingroup$ Actually, the Caferelli et al paper is available via researchgate (google-scholar it and check "all 4 versions"). $\endgroup$
    – Dirk
    Nov 7, 2016 at 16:23
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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

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