Abstract

We continue research on a certain cosine function defined for smooth Minkowski spaces. We prove that such function is symmetric if and only if the corresponding space is Euclidean, and also that it can be given in terms of the Gateaux derivative of the norm. As an application we study the ratio between the lengths of tangent segments drawn from an external point to the unit circle of a Radon plane. We also give a characterization of such planes in terms of signs of the cosine function.

Author information

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

Vitor Balestro

CEFET/RJ Campus Nova Friburgo, 28635-000 Nova Friburgo, Brazil
and: Inst. de Matemática e Estatística, Universidade Federal Fluminense, 24020-140 Niterói, Brazil

vitorbalestro@id.uff.br

Emad Shonoda

Dept. of Mathematics and Computer Science, Faculty of Science, Port Said University, 42521 Port Said, Egypt

en_shonoda@yahoo.de

Keywords

  • Gateaux derivative
  • Minkowski cosine function
  • Minkowski geometry
  • Radon curves
  • semi-inner product
  • smooth norm

Mathematics Subject Classification

46B20; 33B10, 52A10, 52A21

References

  1. J. Alonso, H. Martini, S. Wu: On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces, Aequationes Math. 83(1-2) (2012) 153–189.https://doi.org/10.1007/s00010-011-0092-zGoogle Scholar
  2. 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
  3. V. Balestro, H. Martini, R. Teixeira: Geometric properties of a sine function extendable to arbitrary normed planes, Monatshefte Math. 181(4) (2017) 781–800.https://doi.org/10.1007/s00605-016-0916-yGoogle Scholar
  4. V. Balestro, H. Martini, R. Teixeira: A new construction of Radon curves and related topics, Aequationes Math. 90(5) (2016) 1013–1024.https://doi.org/10.1007/s00010-016-0423-1Google Scholar
  5. V. Balestro, H. Martini, R. Teixeira: Extremal values for constants in normed planes, preprint, available at: https://arxiv.org/abs/1602.06741 (2016).Google Scholar
  6. W. Barthel: Zur Minkowski-Geometrie, begründet auf dem Flächeninhaltsbegriff, Monatshefte Math. 63(4) (1959) 317–343.https://doi.org/10.1007/BF01299548Google Scholar
  7. W. Blaschke: Räumliche Variationsprobleme mit symmetrischer Transversalitätsbedingung, Ber. Verh. Sächs. Ges. Wiss. Leipzig, Math.-Phys. Kl. 68 (1916) 50–55.Google Scholar
  8. H. Busemann: The foundations of Minkowskian geometry, Comment. Math. Helvet. 24(1) (1950) 156–187.https://doi.org/10.1007/BF02567031Google Scholar
  9. S. S. Dragomir: Semi-inner Products and Applications, Nova Science Publishers, Hauppauge (2004).Google Scholar
  10. N. Düvelmeyer: A new characterization of Radon curves via angular bisectors, J. Geometry 80(1) (2004) 75–81.https://doi.org/10.1007/s00022-003-1717-8Google Scholar
  11. P. Finsler: Über eine Verallgemeinerung des Satzes von Meusnier, Vierteljahresschr. Naturforsch. Ges. Zürich 85 (1940) 155–164.Google Scholar
  12. H. Guggenheimer: Pseudo-Minkowski differential geometry, Ann. Mat. Pura Appl.(4), 70(1) (1965) 305–370.https://doi.org/10.1007/BF02410096Google Scholar
  13. H. Guggenheimer: On plane Minkowski geometry, Geom. Dedicata 12(4) (1982) 371–381.https://doi.org/10.1007/BF00147579Google Scholar
  14. H. Martini, K. J. Swanepoel: The geometry of Minkowski spaces – a survey. Part II, Expositiones Math. 22(2) (2004) 93–144.https://doi.org/10.1016/S0723-0869(04)80009-4Google Scholar
  15. H. Martini, K. J. Swanepoel: Antinorms and Radon curves, Aequationes Math. 72(1-2) (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(2) (2001) 97–142.https://doi.org/10.1016/S0723-0869(01)80025-6Google Scholar
  17. P. M. Miličič: Sur le produit scalaire géneralisé, Mat. Vesnik 25(10) (1973) 325–329.Google Scholar
  18. P. M. Miličič: Sur la g-ortogonalité dans des espaces normés, Mat. Vesnik 39 (1987) 325–334.Google Scholar
  19. P. M. Miličič: Une généralisation naturelle du produit scalaire dans un espace normé et son utilisation, Publ. de l’Inst. Math (Beograd) 42(56) (1987) 63–70.Google Scholar
  20. P. M. Miličič: La fonctionelle g et quelques problémes des meilleures approximations dans des espaces normés, Publ. de l’Inst. Math (Beograd) 48(62) (1990) 110–118.Google Scholar
  21. P. M. Miličič: On the B-angle and g-angle in normed spaces, J. Inequal. Pure Appl. Math 8(3) (2007) 1–9.https://doi.org/10.1155/2007/70597Google Scholar
  22. P. Petersen: Riemannian Geometry, second edition, Springer-Verlag, Berlin (2006).Google Scholar
  23. C. M. Petty: On the geometry of the Minkowski plane, Riv. Mat. Univ. Parma 6 (1955) 269–292.Google Scholar
  24. C. M. Petty, J. E. Barry: A geometrical approach to the second-order linear differential equation, Canad. J. Math. 14(2) (1962) 349.https://doi.org/10.4153/CJM-1962-027-xGoogle Scholar
  25. E. Shonoda: On Ruled Surfaces in Three-dimensional Minkowski Space, Lambert Academic Publishing (2011).Google Scholar
  26. E. Shonoda, G. Weiss: Brauner’s angle formula and the theory of curves in Minkowski planes and spaces, in: 16th International Conference on Geometry and Graphics, Innsbruck, Austria, August (2014).Google Scholar
  27. 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
  28. A. C. Thompson: Minkowski Geometry, Encyclopedia Math. Appl. 63, Cambridge University Press, Cambridge (1996).https://doi.org/10.1017/CBO9781107325845Google Scholar
  29. J. Valentine, S. Wayment: Wilson angles in linear normed spaces, Pacific J. Math. 36(1) (1971) 239–243.https://doi.org/10.2140/pjm.1971.36.239Google Scholar
  30. S. Wu: Tangent segments in Minkowski planes, Beiträge zur Algebra und Geometrie 49(1) (2008) 147–151.Google Scholar

Suggested citation

V. Balestro, E. Shonoda. “On a Cosine Function Defined for Smooth Normed Spaces.” Journal of Convex Analysis 25 (2018), No. 1, 21–39. https://doi.org/10.68381/jca25002

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Published by Heldermann Verlag, 2018. Rights now held by Banach Press.