The attainment set of the $\varphi$-envelope and genericity properties

  • A. Cabot
  • A. Jourani
  • L. Thibault
  • D. Zagrodny


The attainment set of the $\varphi$-envelope of a function at a given point is investigated. The inclusion of the attainment set of the $\varphi$-envelope of the closed convex hull of a function into the attainment set of the function is preserved in sufficiently general settings to encompass the case $\varphi$ being a norm in a power not less than $1$. The non-emptiness of the attainment set is guaranteed on generic subsets of a given space, in several fundamental cases.


Asplund, E., Sets with unique farthest points, Israel J. Math. 5 (1967), 201–209.

Asplund, E. and Rockafellar, R. T., Gradients of convex functions, Trans. Amer. Math. Soc. 139 (1969), 443–467.

Bauschke, H. H., Macklem, M. S., and Wang, X., Chebyshev sets, Klee sets, and Chebyshev centers with respect to Bregman distances: recent results and open problems, in “Fixed-point algorithms for inverse problems in science and engineering”, Springer Optim. Appl., vol. 49, Springer, New York, 2011, pp. 1--21.

Borwein, J. M. and Vanderwerff, J. D., Convex functions: constructions, characterizations and counterexamples, Encyclopedia of Mathematics and its Applications, vol. 109, Cambridge University Press, Cambridge, 2010.

Butnariu, D. and Iusem, A. N., Totally convex functions for fixed points computation and infinite dimensional optimization, Applied Optimization, vol. 40, Kluwer Academic Publishers, Dordrecht, 2000.

Cabot, A., Jourani, A., and Thibault, L., Envelopes for sets and functions: regularization and generalized conjugacy, Mathematika 63 (2017), no. 2, 383–432.

Cibulka, R. and Fabian, M., Attainment and (sub)differentiability of the supremal convolution of a function and square of the norm, J. Math. Anal. Appl. 393 (2012), no. 2, 632–643.

Deville, R., Godefroy, G., and Zizler, V., Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific & Technical, Harlow, 1993.

Edelstein, M., Farthest points of sets in uniformly convex Banach spaces, Israel J. Math. 4 (1966), 171–176.

Hiriart-Urruty, J.-B., La conjecture des points les plus éloignés revisitée, Ann. Sci. Math. Québec 29 (2005), no. 2, 197–214.

Ivanov, G. E., Farthest points and the strong convexity of sets, Math. Notes 87 (2010), no. 3--4, 355–366.

Jourani, A., Thibault, L., and Zagrodny, D., The NSLUC property and Klee envelope, Math. Ann. 365 (2016), no. 3-4, 923–967.

Klee, V., Convexity of Chevyshev sets, Math. Ann. 142 (1960/1961), 292–304.

Lau, K. S., Farthest points in weakly compact sets, Israel J. Math. 22 (1975), no. 2, 168–174.

Moreau, J. J., Fonctionnelles convexes, Séminaire Jean Leray 1966–1967 (1967), no. 2, 1–108; second edition, Università di Roma “Tor Vergata”, Dipartimento di Ingegneria Civile, 2003.

Rudin, W., Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York, 1973.

Wang, X., On Chebyshev functions and Klee functions, J. Math. Anal. Appl. 368 (2010), no. 1, 293–310.
How to Cite
Cabot, A., Jourani, A., Thibault, L., & Zagrodny, D. (2019). The attainment set of the $\varphi$-envelope and genericity properties. MATHEMATICA SCANDINAVICA, 124(2), 203-246.