Duality of gauges and symplectic forms in vector spaces

V. Balestro, H. Martini and R. Teixeira, Duality of gauges and symplectic forms in vector spaces, Preprint, 2019. (arXiv link)

Comments: A gauge in a vector space is given by a convex body containing the origin as an interior point. We use the identification between an even-dimensional vector field and its dual space given by a fixed symplectic form to introduce a (skew-)dual gauge.  This is the gauge whose unit ball is the image of the polar body under the described identification. As a consequence of this construction, we prove that closed characteristics in the boundary of a smooth convex body are optimal cases of an isoperimetric inequality which involves symplectic area and length measured in the dual gauge.