# Duality of gauges and symplectic forms in vector spaces

## TL;DR

This paper explores the duality between gauges and symplectic forms in vector spaces, extending the concept of Banach spaces by removing symmetry, and investigates geometric properties and applications such as isoperimetric inequalities.

## Contribution

It introduces the concept of dual gauges induced by symplectic forms and studies their geometric properties, including isometries and orthogonality, extending gauge theory beyond symmetric spaces.

## Key findings

- Dual gauges exhibit specific behaviors under isometries.
- A version of the Mazur-Ulam theorem for gauges is established.
- Closed characteristics of convex bodies relate to isoperimetric inequalities.

## 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.