Duality of gauges and symplectic forms in vector spaces

A gauge γ 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 , γ ) 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 ∘ ⊆ 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.