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Journal of Convex Analysis 25 (2018), No. 1, 021--039
Copyright Heldermann Verlag 2018

On a Cosine Function Defined for Smooth Normed Spaces

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

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

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.

Keywords: Gateaux derivative, Minkowski cosine function, Minkowski geometry, Radon curves, semi-inner product, smooth norm.

MSC: 46B20; 33B10, 52A10, 52A21

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