Abstract

We define new geometric constants for normed planes, determine their optimal values, and characterize types of planes for which these optimal values are attained. Relations of these constants to several topics, such as areas and distances from points to sides of triangles, are also investigated. We perform some calculations of new and old constants given by trigonometric functions for certain classes of norms, and a conjecture on a certain geometric constant is disproved.

Author information

Contact details are reproduced from the original publication and may be historical.

Ralph Teixeira

Instituto de Matemática e Estatística, UFF, Niterói, Brazil

ralph@mat.uff.br

Keywords

  • Banach spaces
  • geometric constants
  • Minkowski geometry
  • Radon norms
  • sine function

Mathematics Subject Classification

46B20; 33B10, 52A10, 52A21

References

  1. J. Alonso, P. Martin, P. L. Papini: Wheeling around von Neumann-Jordan constant in Banach spaces, Studia Math. 188 (2008) 135–150.https://doi.org/10.4064/sm188-2-3Google Scholar
  2. J. Alonso, H. Martini, S. Wu: On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces, Aequationes Math. 83 (2012) 153–189.https://doi.org/10.1007/s00010-011-0092-zGoogle Scholar
  3. V. Balestro, H. Martini, R. Teixeira: Geometric constants for quantifying the difference between orthogonality types, Ann. Funct. Anal. 7(4) (2016) 656–671.https://doi.org/10.1215/20088752-3661053Google Scholar
  4. V. Balestro, H. Martini, R. Teixeira: Geometric properties of a sine function extendable to arbitrary normed planes, Monatsh. Math. 182(4) (2017) 781–800.https://doi.org/10.1007/s00605-016-0916-yGoogle Scholar
  5. J. A. Clarkson: The von Neumann-Jordan constant for the Lebesgue space, Ann. of Math. 38 (1937) 114–115.https://doi.org/10.2307/1968512Google Scholar
  6. J. A. Clarkson: Uniformly convex spaces, Trans. Amer. Math. Soc. 40(3) (1936) 396–414.https://doi.org/10.1090/S0002-9947-1936-1501880-4Google Scholar
  7. J. Gao, K. S. Lau: On two classes of Banach spaces with uniform normal structure, Studia Math. 99 (1991) 41–56.https://doi.org/10.4064/sm-99-1-41-56Google Scholar
  8. E. Heil: Abschätzungen für einige Affininvarianten konvexer Kurven, Monatsh. Math. 71 (1967) 405–423.https://doi.org/10.1007/BF01295132Google Scholar
  9. D. Ji, S. Wu: Quantitative characterization of the difference between Birkhoff orthogonality and isosceles orthogonality, J. Math. Anal. Appl. 323(1) (2006) 1–7.https://doi.org/10.1016/j.jmaa.2005.10.004Google Scholar
  10. R. C. James: Uniformly non-square Banach spaces, Ann. of Math. 80 (1964) 542–550.https://doi.org/10.2307/1970663Google Scholar
  11. M. Kato, L. Maligranda, Y. Takahashi: On James, Jordan-von Neumann constants and the normal structure coefficients of Banach spaces, Studia Math. 144 (2001) 275–295.https://doi.org/10.4064/sm144-3-5Google Scholar
  12. M. Lassak: Affine regular hexagons of extreme areas inscribed in a centrally symmetric convex body, Adv. Geom. 3 (2003) 45–51.https://doi.org/10.1515/advg.2003.004Google Scholar
  13. L. Maligranda, L. I. Nikolova, L. E. Persson, T. Zachariades: On n-th James and Khinchine constants of Banach spaces, Math. Inequal. Appl. 11 (2008) 1–22.https://doi.org/10.7153/mia-11-01Google Scholar
  14. H. Martini, K. Swanepoel: Equiframed curves – a generalization of Radon curves, Monatsh. Math. 141 (2004) 301–314.https://doi.org/10.1007/s00605-003-0052-3Google Scholar
  15. H. Martini, K. J. Swanepoel: Antinorms and Radon curves, Aequationes Math. 71 (2006) 110–138.https://doi.org/10.1007/s00010-006-2825-yGoogle Scholar
  16. H. Martini, K. J. Swanepoel, G. Weiss: The geometry of Minkowski spaces – a survey. Part I, Expositiones Math. 19 (2001) 97–142.https://doi.org/10.1016/S0723-0869(01)80025-6Google Scholar
  17. H. Martini, K. J. Swanepoel: The geometry of Minkowski spaces – a survey. Part II, Expositiones Math. 22 (2004) 93–144.https://doi.org/10.1016/S0723-0869(04)80009-4Google Scholar
  18. L. Y. Nikolova, L. E. Persson, T. Zachariades: A study of some constants for Banach spaces, C. R. Acad. Bulgare Sci. 57 (2004) 5–8.Google Scholar
  19. P. L. Papini, S. Wu: Measurements of differences between orthogonality types, J. Math. Anal. Appl. 397(1) (2013) 285–291.https://doi.org/10.1016/j.jmaa.2012.07.059Google Scholar
  20. T. Szostok: On a generalization of the sine function, Glas. Mat. 38(1) (2003) 29–44.https://doi.org/10.3336/gm.38.1.04Google Scholar
  21. Y. Takahashi: Some geometric constants of Banach spaces, a unified approach, in: Banach and Function Spaces II, M. Kato, L. Maligranda (eds.), Yokohama Publishers, Yokohama (2007) 191–220.Google Scholar
  22. Y. Takahasi, M. Kato: A simple inequality for the von Neumann-Jordan and James constants of a Banach space, J. Math. Anal. Appl. 359 (2009) 602–609.https://doi.org/10.1016/j.jmaa.2009.05.051Google Scholar
  23. A. C. Thompson: Minkowski Geometry, Encyclopedia of Mathematics and its Applications 63, Cambridge University Press, Cambridge (1996).Google Scholar
  24. Z. Zeng: Some results in combinatorial geometry, Dissertation, Universität Bielefeld (1993).Google Scholar

Suggested citation

V. Balestro, H. Martini, R. Teixeira. “Optimal Constants in Normed Planes.” Journal of Convex Analysis 26 (2019), No. 1, 89–104. https://doi.org/10.68381/jca26006

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Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.