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Journal of Convex Analysis 26 (2019), No. 1, 089--104
Copyright Heldermann Verlag 2019

Optimal Constants in Normed Planes

Vitor Balestro
CEFET/RJ, Campus Nova Friburgo, Nova Friburgo, Brazil

Horst Martini
Fakultät für Mathematik, Technische Universität, 09107 Chemnitz, Germany

Ralph Teixeira
Instituto de Matemática e Estatística, UFF, Niterói, Brazil

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.

Keywords: Banach spaces, geometric constants, Minkowski geometry, Radon norms, sine function.

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

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