On a Separation Theorem for Delta-Convex Functions

Olbryś, Andrzej

doi:10.2478/amsil-2020-0013

Abstract

In the present paper we establish necessary and sufficient conditions under which two functions can be separated by a delta-convex function. This separation will be understood as a separation with respect to the partial order generated by the Lorentz cone. An application to a stability problem for delta-convexity is also given.

References

[1] A.D. Aleksandrov, On surfaces represented as the difference of convex functions, Izvestiya Akad. Nauk Kazah. SSR. 60, Ser. Mat. Meh. 3, (1949), 3–20.
Search in Google Scholar Back to article
[2] A.D. Aleksandrov, Surfaces represented by the differences of convex functions, Doklady Akad. Nauk SSSR (N.S.) 72, (1950), 613–616.
Search in Google Scholar Back to article
[3] M.G. Arsove, Functions representable as differences of subharmonic functions, Trans. Amer. Math. Soc. 75 (1953), 327–365.10.1090/S0002-9947-1953-0059416-3
Search in Google Scholar Back to article
[4] K. Baron, J. Matkowski, and K. Nikodem, A sandwich with convexity, Math. Pannon. 5 (1994), no. 1, 139–144.
Search in Google Scholar Back to article
[5] H. Busemann and W. Feller, Krümmungseigenschaften Konvexer Flächen, Acta Math. 66 (1936), no. 1, 1–47.10.1007/BF02546515
Search in Google Scholar Back to article
[6] P. Hartman, On functions representable as a difference of convex functions, Pacific J. Math. 9 (1959), 707–713.10.2140/pjm.1959.9.707
Search in Google Scholar Back to article
[7] C.O. Kiselman, Fonctions delta-convexes, delta-sousharmoniques et delta-plurisousharmoniques, in: P. Lelong (ed.), Séminaire Pierre Lelong (Analyse). Année 1975/76, Lecture Notes in Math., vol. 578, Springer-Verlag, Berlin, 1977, pp. 93–107.10.1007/BFb0091464
Search in Google Scholar Back to article
[8] A. Olbryś, On separation by h-convex functions, Tatra Mt. Math. Publ. 62 (2015), 105–111.10.1515/tmmp-2015-0008
Search in Google Scholar Back to article
[9] A. Olbryś, On sandwich theorem for delta-subadditive and delta-superadditive mappings, Results Math. 72 (2017), no. 1–2, 385–399.10.1007/s00025-016-0627-7
Search in Google Scholar Back to article
[10] A.W. Roberts and D.E. Varberg, Convex Functions, Pure and Applied Mathematics, vol. 57, Academic Press, New York–London, 1973.
Search in Google Scholar Back to article
[11] L. Veselý and L. Zajiček, Delta-convex mappings between Banach spaces and applications, Dissertationes Math. (Rozprawy Mat.) 289 (1989), 52 pp.
Search in Google Scholar Back to article

On a Separation Theorem for Delta-Convex Functions

Abstract

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