Duality of gauges and symplectic forms in vector spaces
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[1] Universidade Federal Fluminense
Universidade Federal Fluminense
Brasil
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[2] Chemnitz University of Technology
Chemnitz University of Technology
Kreisfreie Stadt Chemnitz, Alemania
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- Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 72, Fasc. 3, 2021, págs. 501-525
- Idioma: inglés
- DOI: 10.1007/s13348-020-00297-z
- Texto completo no disponible (Saber más ...)
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Resumen
A gauge \gamma in a vector space X is a distance function given by the Minkowski functional associated to a convex body K containing the origin in its interior. Thus, the outcoming concept of gauge spaces (X, \gamma ) extends that of finite dimensional real Banach spaces by simply neglecting the symmetry axiom (a viewpoint that Minkowski already had in mind). If the dimension of X is even, then the fixation of a symplectic form yields an identification between X and its dual space X^*. The image of the polar body K^{\circ }\subseteq X^* under this identification yields a (skew-)dual gauge on X. In this paper, we study geometric properties of this so-called dual gauge, such as its behavior under isometries and its relation to orthogonality. A version of the Mazur–Ulam theorem for gauges is also proved. As an application of the theory, we show that closed characteristics of the boundary of a (smooth) convex body are optimal cases of a certain isoperimetric inequality.
