Characterizations of hilbertian norms involving the areas of triangles in a smooth space
DOI:
https://doi.org/10.24193/subbmath.2022.1.10Keywords:
Smooth space, strictly convex space, norm derivative, Birkhoff orthogonality, height of a triangle, hilbertian norm.Abstract
In the previous paper, we have defined together with I. Ionica˘ the heights of a nontrivial triangle with respect to Birkhoff orthogonality in a real smooth space X, dim X ≥ 2. In the present paper, we remark that, generally, the area of a nontrivial triangle in X has not the same value for different heights of the triangle. The purpose of this paper is to characterize the norm of X if this space has the property that the area of any triangle is well defined (independent of considered height). In this line we give five equivalent properties using the directional derivative of the norm. If X is strictly convex and dimX ≥ 3, then each of these five properties characterizes the hilbertian norms (generated by inner products).
Mathematics Subject Classification (2010): 54AXX.
Received 21 July 2021; Revised 01 February 2022; Accepted 09 February 2022.
References
Alsina, C., Guijarro, P., Tomas, M.S., On heights in real normed spaces and characterizations of inner product structures, J. Inst. Math. Comput. Sci. Math. Ser., 6(1993), 151-159.
Alsina, C., Guijarro, P., Tomas, M.S., A characterization of inner product space based on a property of height’s transformation, Arch. Mat., 6(1993), 560-566.
Alsina, C., Guijarro, P., Tomas, M.S., Some remarkables lines of triangles in real normed spaces and characterizations of inner product structures, Aequationes Math., 54(1997), 234-241.
Alsina, C., Sikorska, J., Tomas, M.S., Norm Derivative Characterizations of Inner Product Spaces, World Sci. Publish. Co., (2010), 188 pp.
Amir, D., Characterizations of Inner Product Spaces, Birkha¨user Verlag, Basel-Boston-Stuttgart, 1986.
Barbu, V., Precupanu, T., Convexity and Optimization in Banach Spaces, Fourth Edition, Springer, Dordrecht Heidelberg London New York, 2012.
Diestel, I., Geometry of Banach Spaces Selected topics, Lecture Notes in Mathematics, Springer, Berlin, 1978.
Giles, J.R., On a characterization of differentiability of the norm of a normed linear spaces, J. Austral. Math. Soc., 12(1971), 106-114.
Guijarro, P., Tomas, M.S., Characterizations of inner product spaces by geometrical properties of the heights in a triangle, Arch. Math., 73(1999), 64-72.
Ionica, I., Bisectrices of a triangle in a linear normed space and hilbertian norms, An. St. Univ. Iasi, 58(2012), 145-156.
Joichi, J.I., More characterizations of inner product spaces, Proc. Amer. Math. Soc., 19(1968), 1180-1186.
Leduc, M., Caracterizations des espaces euclidiens, C.R. Acad. Sci. Paris, Ser. A-B, 268(1969), 943-946.
Papini, P.L., Inner products and norm derivatives, J. Math. Anal. Appl., 91(1983), 592- 598.
Precupanu, T., Ionica, I., Heights of a triangle in a linear normed space and hilbertian norms, A. St. Univ. ”Al.I. Cuza” Iasi, Matematica, 55(2009), 35-43.
Precupanu, T., Characterizations of hilbertian norms, Bull. U.M.I., 5(1978), no. 15-B, 161-169.
Precupanu, T., On a characterization of hilbertian norms, An. St. Univ. ”Al.I. Cuza” Iasi, Matematica, 56(2010).
Tapia, R.A., A characterizations of inner product spaces, Proc. AMS, 41(1973), 569-574.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2022 Studia Universitatis Babeș-Bolyai Mathematica
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
