Abstract
Separation theorems play a central role in the theory of Functional Inequalities. The importance of Convex Geometry has led to the study of convexity structures induced by Beckenbach families. The aim of the present note is to replace recent investigations into the context of an axiomatic setting, for which Beckenbach structures serve as models. Besides the alternative approach, some new results (whose classical correspondences are well-known in Convex Geometry) are also presented.
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Balaj M., Wasowicz S.: Haar spaces and polynomial selections. Math. Pannon. 14(1), 63–70 (2003)
Baron K., Matkowski J., Nikodem K.: A sandwich with convexity. Math. Pannon. 5(1), 139–144 (1994)
Beckenbach, E.F.: Generalized convex functions, Bull. Amer. Math. Soc. 43(6), 363–371 (1937)
Bessenyei, M., Konkoly, Á., Popovics, B.: Convexity with respect to Beckenbach families, J. Convex Anal. (2015) to appear
Bessenyei, M., Páles, Zs.: Separation by linear interpolation families. J. Nonlinear Convex Anal. 13(1), 49–56 (2012)
Bessenyei M., Szokol, P.: Convex separation by regular pairs, J. Geom. 104(1), 45–56 (2013)
Bessenyei M., Szokol, P.: Separation by convex interpolation families. J. Convex Anal. 20(4), 937–946 (2013)
Carathéodory, C.: \"Uber den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen, Rend. Circ. Mat. Palermo 32, 193–217 (1911)
Clarke, F.H., Ledyaev, Yu.S., Stern, R.J., Wolenski, P.R.: Nonsmooth analysis and control theory, Graduate Texts in Mathematics, vol. 178, Springer, New York (1998)
Conway, J.B.: A course in functional analysis, Graduate Texts in Mathematics, vol. 96, Springer, New York (1985)
Danzer, L., Grünbaum, B., Klee, V.: Helly’s theorem and its relatives, Proc. Sympos. Pure Math., Vol. VII, Amer. Math. Soc., Providence, R.I., pp. 101–180 (1963)
Hartshorne, R.: Geometry: Euclid and beyond, Undergraduate Texts in Mathematics, Springer, New York (2000)
Helly, E.: Über Mengen konvexer Körper mit gemeinschftlichen Punkten, Jahresbericht Deutsch. Math. Vereinigung 32, 175–176 (1923)
Hilbert, D.: The foundations of geometry (1899), The Open Court Publishing Company, University of Illinois (1950)
Kakutani, D.: Ein Beweis des Sätzes von Edelheit über konvexe Mengen, Proc. Imp. Acad. Tokyo 13, 93–94 (1937)
Karlin, S., Studden, W.J.: Tchebycheff systems: With applications in analysis and statistics, Pure and Applied Mathematics, Vol. XV, Interscience Publishers John Wiley & Sons, New York-London-Sydney (1966)
König, D.: Über konvexe Körper, Math. Zeitschrift 14, 208–220 (1922)
Krzyszkowski, J.: Generalized convex sets, Rocznik Nauk.-Dydakt. Prace Mat. (14), 59–68 (1997)
Krzyszkowski, J.: Approximately generalized convex functions, Math. Pannon. 12(1), 93–104 (2001)
Moore, E.H.: On the projective axioms of geometry, Trans. Amer. Math. Soc. 3(1), 142–158 (1902)
Moulton F.R.: A simple non-desarguesian geometry. Trans. Am. Math. Soc. 14(2), 192–195 (1902)
Nikodem, K., Páles, Zs.: Generalized convexity and separation theorems. J. Convex Anal. 14(2), 239–247 (2007)
Nikodem K., Wasowicz, S. A sandwich theorem and Hyers-Ulam stability of affine functions, Aequationes Math. 49(1-2), 160–164 (1995)
Popoviciu, T.: Les fonctions convexes, Actualités Sci. Ind., no. 992, Hermann et Cie, Paris (1944)
Radon, J.: Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten, Math. Ann. 83, 113–115 (1921)
Rådström, H.: One-parameter semigroups of subsets of a real linear space, Ark. Mat., 4, 87–97 (1960)
Rockafellar, R.T.: Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J. (1970)
Stone, M.H.: Convexity, mimeographed lecture notes, University of Chicago (1946)
Tornheim, L.: On n-parameter families of functions and associated convex functions, Trans. Am. Math. Soc. 69, 457–467 (1950)
van de Vel, M.L.J.: Theory of convex structures, North-Holland Mathematical Library, vol. 50, North-Holland Publishing Co., Amsterdam (1993)
Wasowicz S.: Polynomial selections and separation by polynomials. Stud. Math. 120(1), 75–82 (1996)
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This research has been supported by the Hungarian Scientific Research Fund (OTKA) Grants K–111651 and by the SROP-4.2.2.B-15/1/KONV-2015-0001 project. The project has been supported by the European Union, co-financed by the European Social Fund.
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Bessenyei, M., Popovics, B. Convexity without convex combinations. J. Geom. 107, 77–88 (2016). https://doi.org/10.1007/s00022-015-0276-0
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DOI: https://doi.org/10.1007/s00022-015-0276-0