Asymmetry measures for convex distance functions

V. Balestro, H. Martini and R. Teixeira, Asymmetry measures for convex distance functions, Preprint, 2019. (arXiv link)

Comments: A convex distance function (or gauge) in a vector space is the distance given by the Minkowski functional of a (non-necessarily symmetric with respect to the origin) convex body containing the origin as an interior point. We want to define asymmetry measures in gauge planes which, not only make sense from the metric viewpoint, but also are indeed asymmetry measures of the related convex bodies. In particular, the asymmetry measures that we define are invariant under gauge isometries, and also are continuous with respect to the Hausdorff distance in the space of convex bodies containing the origin as an interior point. Also, we prove that two of the measures have a duality principle with respect to the dual gauge given by the fixation of a symplectic form in the plane.