The attainment set of the $\varphi$-envelope and genericity properties
DOI:
https://doi.org/10.7146/math.scand.a-110766Abstract
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.
References
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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. https://doi.org/10.1016/j.jmaa.2012.03.052
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Edelstein, M., Farthest points of sets in uniformly convex Banach spaces, Israel J. Math. 4 (1966), 171–176. https://doi.org/10.1007/BF02760075
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. https://doi.org/10.1134/S0001434610030065
Jourani, A., Thibault, L., and Zagrodny, D., The NSLUC property and Klee envelope, Math. Ann. 365 (2016), no. 3-4, 923–967. https://doi.org/10.1007/s00208-015-1283-z
Klee, V., Convexity of Chevyshev sets, Math. Ann. 142 (1960/1961), 292–304. https://doi.org/10.1007/BF01353420
Lau, K. S., Farthest points in weakly compact sets, Israel J. Math. 22 (1975), no. 2, 168–174. https://doi.org/10.1007/BF02760164
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. https://doi.org/10.1016/j.jmaa.2010.03.041
Asplund, E. and Rockafellar, R. T., Gradients of convex functions, Trans. Amer. Math. Soc. 139 (1969), 443–467. https://doi.org/10.2307/1995335
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. https://doi.org/10.1007/978-1-4419-9569-8_1
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. https://doi.org/10.1017/CBO9781139087322
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. https://doi.org/10.1007/978-94-011-4066-9
Cabot, A., Jourani, A., and Thibault, L., Envelopes for sets and functions: regularization and generalized conjugacy, Mathematika 63 (2017), no. 2, 383–432. https://doi.org/10.1112/S0025579316000309
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. https://doi.org/10.1016/j.jmaa.2012.03.052
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. https://doi.org/10.1007/BF02760075
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. https://doi.org/10.1134/S0001434610030065
Jourani, A., Thibault, L., and Zagrodny, D., The NSLUC property and Klee envelope, Math. Ann. 365 (2016), no. 3-4, 923–967. https://doi.org/10.1007/s00208-015-1283-z
Klee, V., Convexity of Chevyshev sets, Math. Ann. 142 (1960/1961), 292–304. https://doi.org/10.1007/BF01353420
Lau, K. S., Farthest points in weakly compact sets, Israel J. Math. 22 (1975), no. 2, 168–174. https://doi.org/10.1007/BF02760164
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. https://doi.org/10.1016/j.jmaa.2010.03.041
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Published
2019-06-17
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. https://doi.org/10.7146/math.scand.a-110766
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