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Duality of gauges and symplectic forms in vector spaces

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Abstract

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.

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Authors and Affiliations

  1. Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, 24210201, Brazil

    Vitor Balestro & Ralph Teixeira

  2. Fakultät für Mathematik, Technische Universität Chemnitz, 09107, Chemnitz, Germany

    Horst Martini

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  1. Vitor Balestro
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  2. Horst Martini
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  3. Ralph Teixeira
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Correspondence to Vitor Balestro.

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Balestro, V., Martini, H. & Teixeira, R. Duality of gauges and symplectic forms in vector spaces. Collect. Math. 72, 501–525 (2021). https://doi.org/10.1007/s13348-020-00297-z

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  • DOI: https://doi.org/10.1007/s13348-020-00297-z

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