Journal of Convex Analysis 27 (2020), No. 1, 117--138
Copyright Heldermann Verlag 2020
Asymmetry Measures for Convex Distance Functions
Vitor Balestro
Inst. de Matemática e Estatística, Universidade Federal Fluminense, 24210201 Niterói, Brazil
vitorbalestro@id.uff.br
Horst Martini
Fakultät für Mathematik, Technische Universität, 09107 Chemnitz, Germany
martini@mathematik.tu-chemnitz.de
Ralph Teixeira
Inst. de Matemática e Estatística, Universidade Federal Fluminense, 24210201 Niterói, Brazil
ralph@mat.uff.br
Gauges, or convex distance functions are, roughly speaking, norms without symmetry. In this paper we intend to quantify how asymmetric a planar gauge can be. We introduce asymmetry measures for smooth gauges and for strictly convex gauges, prove that they are invariant under isometries, and investigate lower and upper bounds for them. Identifying a gauge with a convex body containing the origin in its interior (the unit ball of the gauge), we also prove that all introduced asymmetry measures are continuous in the Hausdorff distance. Finally, we show that, modifying one of the constructed asymmetry measures, a certain duality principle holds.
Keywords: Asymmetry measure, convex distance function, gauge space, Hausdorff distance, Mazur-Ulam theorem, orthogonality, symplectic form.
MSC: 52A10, 52A20, 52A21, 52A27, 52A38, 46B20.
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