Abstract
In the present paper, we introduce a new concept of convexity which is generated by a family of endomorphisms of an Abelian group. In Abelian groups, equipped with a translation invariant metric, we define the boundedness, the norm, the modulus of injectivity and the spectral radius of endomorphisms. Beyond the investigation of their properties, our first main goal is an extension of the celebrated Rådström cancellation theorem. Another result generalizes the Neumann invertibility theorem. Next we define the convexity of sets with respect to a family of endomorphisms, and we describe the set-theoretical and algebraic structure of the class of such sets. Given a subset, we also consider the family of endomorphisms that make this subset convex, and we establish the basic properties of this family. Our first main result establishes conditions which imply midpoint convexity. The next main result, using our extension of the Rådström cancellation theorem, presents further structural properties of the family of endomorphisms that make a subset convex.
Funding statement: The research of the second author was supported by the 2019-2.1.11-TÉT-2019-00049, EFOP-3.6.1-16-2016-00022 and EFOP-3.6.2-16-2017-00015 projects. The last two projects are co-financed by the European Union and the European Social Fund.
References
[1] S. J. Dilworth, R. Howard and J. W. Roberts, Extremal approximately convex functions and estimating the size of convex hulls, Adv. Math. 148 (1999), no. 1, 1–43. 10.1006/aima.1999.1836Search in Google Scholar
[2] S. J. Dilworth, R. Howard and J. W. Roberts, On the size of approximately convex sets in normed spaces, Studia Math. 140 (2000), no. 3, 213–241. 10.4064/sm-140-3-213-241Search in Google Scholar
[3] S. J. Dilworth, R. Howard and J. W. Roberts, Extremal approximately convex functions and the best constants in a theorem of Hyers and Ulam, Adv. Math. 172 (2002), no. 1, 1–14. 10.1006/aima.2001.2058Search in Google Scholar
[4] R. B. Holmes, A Course on Optimization and best Approximation, Lecture Notes in Math. 257, Springer, Berlin, 1972. 10.1007/BFb0059450Search in Google Scholar
[5] W. Jarczyk and Z. Páles, Convexity and a Stone-type theorem for convex sets in abelian semigroup setting, Semigroup Forum 90 (2015), no. 1, 207–219. 10.1007/s00233-014-9613-0Search in Google Scholar
[6] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Uniwersytet Śla̧ski, Katowice, 1985. Search in Google Scholar
[7] N. Kuhn, A note on t-convex functions, General Inequalities. 4 (Oberwolfach 1983), Internat. Schriftenreihe Numer. Math. 71, Birkhäuser, Basel (1984), 269–276. 10.1007/978-3-0348-6259-2_25Search in Google Scholar
[8] A. Lelek and W. Nitka, On convex metric spaces. I, Fund. Math. 49 (1960/61), 183–204. 10.4064/fm-49-2-183-204Search in Google Scholar
[9] K. Menger, Untersuchungen über allgemeine Metrik, Math. Ann. 100 (1928), no. 1, 75–163. 10.1007/978-3-7091-6110-4_20Search in Google Scholar
[10] Z. Páles, A Hahn–Banach theorem for separation of semigroups and its applications, Aequationes Math. 37 (1989), no. 2–3, 141–161. 10.1007/BF01836441Search in Google Scholar
[11] Z. Páles, A Stone-type theorem for abelian semigroups, Arch. Math. (Basel) 52 (1989), no. 3, 265–268. 10.1007/BF01194389Search in Google Scholar
[12] H. Rå dström, An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3 (1952), 165–169. 10.1090/S0002-9939-1952-0045938-2Search in Google Scholar
[13] H. Rå dström, Convexity and norm in topological groups, Ark. Mat. 2 (1952), 99–137. 10.1007/BF02590879Search in Google Scholar
[14] H. Reiter, Convex sets, Cantor sets and a midpoint property, Canad. Math. Bull. 19 (1976), no. 4, 467–471. 10.4153/CMB-1976-070-9Search in Google Scholar
[15] M. H. Stone, Convexity, University of Chicago, Chicago, 1946. Search in Google Scholar
[16] M. L. J. van de Vel, Theory of Convex Structures, North-Holland Math. Libr. 50, North-Holland, Amsterdam, 1993. Search in Google Scholar
[17] Z. Zhang and Z. Shi, Convexities and approximative compactness and continuity of metric projection in Banach spaces, J. Approx. Theory 161 (2009), no. 2, 802–812. 10.1016/j.jat.2009.01.003Search in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
