On the smoothness of convex envelopes

Griewank, A.; Rabier, P. J.

doi:10.1090/S0002-9947-1990-0986024-2

On the smoothness of convex envelopes
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by A. Griewank and P. J. Rabier

Trans. Amer. Math. Soc. 322 (1990), 691-709

DOI: https://doi.org/10.1090/S0002-9947-1990-0986024-2

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Abstract:

We examine differentiability properties of the convex envelope $\operatorname {conv} E$ of a given function $E:{{\mathbf {R}}^n} \to ( - \infty ,\infty ]$ in terms of properties of $E$. It is shown that ${C^1}$ as well as optimal ${C^{1,\alpha }}$ regularity results, $0 < \alpha \leqslant 1$, can be obtained under general conditions.

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