Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

Some Geometric Constants Related to the Heights and Midlines of Triangles in Banach Spaces

  • Dandan Du (Department of Mathematics, Sun Yat-sen University) ;
  • Yuankang Fu (Department of Mathematics, Sun Yat-sen University) ;
  • Zhijian Yang (Department of Mathematics, Sun Yat-sen University) ;
  • Yongjin Li (Department of Mathematics, Sun Yat-sen University)
  • Received : 2022.04.19
  • Accepted : 2022.10.07
  • Published : 2023.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we introduce two new geometric constants related to the heights of triangles: ∆H(X) and ∆h(X, I). We also propose two new geometric constants, ∆m(X) and ∆M(X), related to the midlines of equilateral triangles, and discuss the relation between the heights and midlines in equilateral triangles. We give estimates for these geometric constants in terms of other geometric parameters, and the geometric constants are used to discuss geometric properties such as uniform non-squareness, uniform normal structure, and the fixed point property.

Keywords

Acknowledgement

This work was supported by the National Natural Science Foundation of P. R. China (Nos.11971493 and 12071491).

References

  1. C. Alsina, P. Guijarro and M. Tomas, Some remarkable lines of triangles in real normed spaces and characterizations of inner product, Aequationes Math., 54(1997), 234-241. https://doi.org/10.1007/BF02755458
  2. J. Bana's, On moduli of smoothness of Banach spaces, Bull. Polish Acad. Sci. Math., 34(1986), 287-293.
  3. D. B'arcenas, V. Gurariy, L. S'anchez, and A. Ull'an, On moduli of convexity in Banach Spaces, Quaest. Math., 27(2004), 137-145. https://doi.org/10.2989/16073600409486089
  4. M. Baronti and P. L. Papini, Convexity, smoothness and moduli, Nonlinear Anal., 70(2009), 2457-2465. https://doi.org/10.1016/j.na.2008.03.030
  5. G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. J., 2(1)(1935), 169-172. https://doi.org/10.1215/S0012-7094-35-00115-6
  6. M. Brodskii and D. Milman, On the center of a convex set, Proc. Amer. Math. Soc., 59(1948), 837-840.
  7. J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc., 40(3)(1936), 396-414. https://doi.org/10.1090/S0002-9947-1936-1501880-4
  8. J. A. Clarkson, The von Neumann-Jordan constant for the Lebesgue space, Ann. of Math., 38(1937), 114-115. https://doi.org/10.2307/1968512
  9. M. M. Day, Uniform convexity in factor and conjugate spaces, Ann. of Math., 45(1944), 375-385. https://doi.org/10.2307/1969275
  10. S. Dhompongsa, A. Kaewkhao and S. Tasena, On a generalized James constant, J. Math. Anal. Appl., 285(2)(2003), 419-435. https://doi.org/10.1016/S0022-247X(03)00408-6
  11. J. Gao, The uniform degree of the unit ball of a Banach spaces(I), Nanjing Daxue Xuebao. Ziran Kexue Ban, 1(1982), 14-28(in Chinese, English summary).
  12. J. Garc'la-Falset, E. Llorens-Fuster and E.M. Mazcu˜nan-Navarro, Uniformly nonsqure Banach spaces have the fixed point property for nonexpansive mappings, J. Funct. Anal., 233(2)(2006), 494-514. https://doi.org/10.1016/j.jfa.2005.09.002
  13. K. Goebal, Convexity of balls and fixed-point theorems for mappings with nonexpansive square, Compos. Math., 22(3)(1970), 269-274.
  14. V. I. Gurariy, On moduli of convexity and flattening of Banach spaces, Soviet Math. Dokl., 161(5)(1965), 1003-1006. (English translation)
  15. V. I. Gurariy and Y. U. Sozonov, Normed spaces that do not have distortion of the unit sphere, Mat. Zametki, 7(1970), 307-310. (Russian)
  16. C. He and Y. A. Cui, Some properties concerning Milman's moduli, J. Math. Anal. Appl., 340(2007), 1260-1272. https://doi.org/10.1016/j.jmaa.2006.07.046
  17. R. C. James, Uniformly non-square Banach space, Ann. of Math., 80(1964), 542-550. https://doi.org/10.2307/1970663
  18. R. C. James, Orthogonality in normed linear spaces, Duke Math. J., 12(1945), 291-302. https://doi.org/10.1215/S0012-7094-45-01223-3
  19. D. H. Ji and S. L. Wu, Quantitative characterization of the difference between Birkhoff orthogonality and isosceles orthogonality, J. Math. Anal. Appl., 323(2006), 1-7. https://doi.org/10.1016/j.jmaa.2005.10.004
  20. M. Kato, L. Maligranda and Y. Takahashi. On James and Jordan-von Neumann constants and the normal structure coefficient of Banach spaces, Stud. Math., 144(3)(2000), 275-295. https://doi.org/10.4064/sm144-3-5
  21. N. Komuro, K. S. Saito and R. Tanaka, On the class of Banach spaces with James constant $\sqrt2$, Math. Nachr., 289(8-9)(2016), 1005-1020. https://doi.org/10.1002/mana.201500238
  22. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II: Function Spaces. 1st ed, Springer: Berlin, Germany(1979).
  23. H. Martini, K. J. Swanepoel and G. Weiss, The geometry of Minkowski spaces- a survey, I, Expo. Math., 19(2)(2001), 97-142. https://doi.org/10.1016/S0723-0869(01)80025-6
  24. V. D. Milman and G. Schechtman, Asymptotic theory of finite-dimensional normed spaces, Springer: Berlin, Germany(1986).
  25. H. Mizuguchi, The constants to measure the differences between Birkhoff and isosceles orthogonalities, Filomat, 30(10)(2016), 2761-2770. https://doi.org/10.2298/FIL1610761M
  26. H. Mizuguchi, The von Neumann-Jordan and another constants in Radon plane, Monatsh. Math., 195(2021), 307-322.  https://doi.org/10.1007/s00605-021-01540-w
  27. B. Z. Ni, C. He and D. H. Ji, A New Characterization of Uniformly Nonsquare Banach Spaces, Journal of Harbin University of Science and Techonlogy, 16(1)(2009), 107-109.
  28. G. Nordlander, The modulus of convexity in normed linear spaces, Ark. Mat., 4(1)(1960), 15-17. https://doi.org/10.1007/BF02591317
  29. P. L. Papini and S. L. Wu, Measurements of differences between orthogonality types, J. Math. Anal. Appl., 397(2013), 285-291. https://doi.org/10.1016/j.jmaa.2012.07.059
  30. G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge University Press, Cambridge, UK(1980).
  31. Y. Takahashi and M. Kato, Von Neumann-Jordan constant and uniformly non-square Banach spaces, Nihonkai Math. J., 9(1998), 155-169.