Skip to main content Accessibility help


  • A. Cabot (a1), A. Jourani (a2) and L. Thibault (a3) (a4)


Let $X$ be a vector space and let $\unicode[STIX]{x1D711}:X\rightarrow \mathbb{R}\cup \{-\infty ,+\infty \}$ be an extended real-valued function. For every function $f:X\rightarrow \mathbb{R}\cup \{-\infty ,+\infty \}$ , let us define the $\unicode[STIX]{x1D711}$ -envelope of $f$ by

$$\begin{eqnarray}f^{\unicode[STIX]{x1D711}}(x)=\sup _{y\in X}\unicode[STIX]{x1D711}(x-y)\begin{array}{@{}c@{}}-\\ \cdot \end{array}f(y),\end{eqnarray}$$
where $\begin{array}{@{}c@{}}-\\ \\ \\ \cdot \end{array}$ denotes the lower subtraction in $\mathbb{R}\cup \{-\infty ,+\infty \}$ . The main purpose of this paper is to study in great detail the properties of the important generalized conjugation map $f\mapsto f^{\unicode[STIX]{x1D711}}$ . When the function $\unicode[STIX]{x1D711}$ is closed and convex, $\unicode[STIX]{x1D711}$ -envelopes can be expressed as Legendre–Fenchel conjugates. By particularizing with $\unicode[STIX]{x1D711}=(1/p\unicode[STIX]{x1D706})\Vert \cdot \Vert ^{p}$ , for $\unicode[STIX]{x1D706}>0$ and $p\geqslant 1$ , this allows us to derive new expressions of the Klee envelopes with index $\unicode[STIX]{x1D706}$ and power $p$ . Links between $\unicode[STIX]{x1D711}$ -envelopes and Legendre–Fenchel conjugates are also explored when $-\unicode[STIX]{x1D711}$ is closed and convex. The case of Moreau envelopes is examined as a particular case. In addition to the $\unicode[STIX]{x1D711}$ -envelopes of functions, a parallel notion of envelope is introduced for subsets of $X$ . Given subsets $\unicode[STIX]{x1D6EC}$ , $C\subset X$ , we define the $\unicode[STIX]{x1D6EC}$ -envelope of $C$ as $C^{\unicode[STIX]{x1D6EC}}=\bigcap _{x\in C}(x+\unicode[STIX]{x1D6EC})$ . Connections between the transform $C\mapsto C^{\unicode[STIX]{x1D6EC}}$ and the aforestated $\unicode[STIX]{x1D711}$ -conjugation are investigated.



Hide All
1. Attouch, H., Buttazzo, G. and Michaille, G., Variational analysis in Sobolev and BV spaces. In Applications to PDE’s and Optimization (MPS/SIAM Series on Optimization 6 ), Society for Industrial and Applied Mathematics (SIAM) (Philadelphia, PA, 2006).
2. Auslender, A. and Teboulle, M., Asymptotic Cones and Functions in Optimization and Variational Inequalities (Springer Monographs in Mathematics), Springer (New York, 2003).
3. Aze, D. and Volle, M., Various continuity properties of the deconvolution. In Advances in Optimization (Lectures Notes in Economics and Mathematical Systems 382 ), Springer (1992), 1630.
4. Balder, E. J., An extension of duality-stability relations to nonconvex optimization problems. SIAM J. Control Optim. 15 1977, 329343.
5. Brønsted, A. and Rockafellar, R. T., On the subdifferentiability of convex functions. Proc. Amer. Math. Soc. 16 1965, 605611.
6. Dolecki, S., Polarities and generalized extremal convolutions. J. Convex Anal. 23 2016, 603614.
7. Dolecki, S. and Kurcyusz, S., On 𝛷-convexity in extremal problems. SIAM J. Control Optim. 16 1978, 277300.
8. Ekeland, I. and Temam, R., Convex analysis and variational problems. In SIAM Classics in Applied Mathematics (EkeTem 28 ), Society for Industrial and Applied Mathematics (Philadelphia, PA, 1999).
9. Elster, K.-H. and Wolf, A., Recent results on generalized conjugate functions. In Trends in Mathematical Optimization, Birkhäuser (Basel, 1988), 6778.
10. Granero, A. S., Jiménez-Sevilla, M. and Moreno, J. P., Intersections of closed balls and geometry of Banach spaces. Extracta Math. 19(1) 2004, 5592.
11. Hiriart-Urruty, J.-B., A general formula on the conjugate of the difference of functions. Canad. Math. Bull. 29 1986, 482485.
12. Hiriart-Urruty, J.-B., The deconvolution operation in convex analysis: an introduction. Cybernet. Systems Anal. 30 1994, 555560.
13. Hiriart-Urruty, J.-B. and Lemaréchal, C., Convex analysis and minimization algorithms, I. Fundamentals, II. In Advanced Theory and Bundle Methods, Springer (Berlin, 1993).
14. Hiriart-Urruty, J.-B. and Mazure, M.-L., Formulations variationnelles de l’addition parallèle et de la soustraction parallèlle d’opérateurs semi-définis positifs. C. R. Acad. Sci. Paris, Sér. I 302 1986, 527530.
15. Ivanov, G. E., Weak convexity of functions and the infimal convolution. J. Convex Anal. 23 2016, 719732.
16. Jourani, A., Thibault, L. and Zagrodny, D., The NSLUC property and Klee envelope. Math. Ann. 365(3–4) 2016, 923967.
17. Martinez-Legaz, J.-E., Generalized conjugation and related topics. In ‘Generalized Convexity and Fractional Programming with Economic Applications’, Proc. Pisa. Italy (1988) (Lecture Notes in Economics and Mathematical Systems 345 ) (ed. Cambini, A. et al. ), Springer (Berlin, 1990), 168197.
18. Martinez-Legaz, J.-E. and Penot, J.-P., Regularization by erasement. Math. Scand. 98 2006, 97124.
19. Mazur, S., Über schwache Konvergentz in den Raumen L p . Studia Math. 4 1933, 128133.
20. Moreau, J. J., Proximité et dualité dans un espace hilbertien. Bull. Soc. Math. France 93 1965, 273299.
21. Moreau, J. J., Inf-convolution, sous-additivité, convexité des fonctions numériques. J. Math. Pures Appl. 49 1970, 109154.
22. Moreau, J. J., Fonctionnelles convexes. In Collège de France, Paris (1967), 2nd edn., Consiglio Nazionale delle Ricerche and Facoltá di Ingegneria Universita di Roma ‘Tor Vergata’ (2003).
23. Penot, J.-P., Calculus Without Derivatives (Graduate Texts in Mathematics), Springer (New York, 2013).
24. Penot, J.-P. and Volle, M., On strongly convex and paraconvex dualities. In ‘Generalized Convexity and Fractional Programming with Economic Applications’, Proc. Pisa. Italy (1988) (Lecture Notes in Economics and Mathematical Systems 345 ) (ed. Cambini, A. et al. ), Springer (Berlin, 1990), 198218.
25. Polovinkin, E. S., On strongly convex sets and strongly convex functions. J. Math. Sci. 100 2000, 26332681.
26. Polovinkin, E. S. and Balashov, M. V., Elements of Convex and Strongly Convex Analysis, Fizmatlit (Moscow, 2004) (Russian).
27. Pshenichnyi, B. N., Leçons sur les jeux différentiels. Cah. l’I.R.I.A.(4) 1971, 145226.
28. Rockafellar, R. T., Convex Analysis, Princeton University Press (Princeton, NJ, 1970).
29. Rockafellar, R. T., Augmented Lagrange multipliers functions and duality in nonconvex programming. SIAM J. Control Optim. 12 1974, 268285.
30. Rockafellar, R. T. and Wets, R. J.-B., Variational Analysis, Springer (Berlin, 1998).
31. Rubinov, A., Abstract Convexity and Global Optimization, Kluwer (Dordrecht, 2000).
32. Singer, I., Conjugation operators. In Selected Topics in Operations Research and Mathematical Economics (eds Hammer, G. and Pallaschke, D.), Springer (Berlin, 1984), 8097.
33. Singer, I., Some Relations between Dualities, Polarities, Coupling Functionals, and Conjugations. J. Math. Anal. Appl. 115 1986, 122.
34. Singer, I., Duality for Nonconvex Approximation and Optimization (CMS Books in Mathematics), Springer (New York, 2006).
35. Vesely, L., Affine mappings and convex functions. Examples of convex functions. Preprint, available at
36. Vial, J.-P., Strong and weak convexity of sets and functions. Math. Oper. Res. 8 1983, 231259.
37. Volle, M., Contributions à la Dualité et à l’Épiconvergence. Thèse de Doctorat d’État, Université de Pau et des Pays de l’Adour, 1986.
38. Volle, M., A formula on the subdifferential of the deconvolution of convex functions. Bull. Aust. Math. Soc. 47 1993, 333340.
39. Wang, X., On Chebyshev functions and Klee functions. J. Math. Anal. Appl. 368 2010, 293310.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed