# Asymmetry measures for convex distance functions

## TL;DR

This paper introduces measures to quantify asymmetry in convex distance functions (gauges), proves their invariance under isometries, explores bounds, and establishes a duality principle, advancing understanding of asymmetric norms.

## Contribution

It defines and analyzes asymmetry measures for convex gauges, proving invariance, continuity, and a duality principle, which are novel contributions to the study of convex distance functions.

## Key findings

- Asymmetry measures are invariant under isometries.
- All asymmetry measures are continuous in the Hausdorff distance.
- A duality principle relates the measures through a modified form.

## Abstract

Gauges, or convex distance functions are, roughly speaking, norms without symmetry. In this paper we intend to quantify how asymmetric a planar gauge can be. We introduce asymmetry measures for smooth gauges and for strictly convex gauges, prove that they are invariant under isometries, and investigate lower and upper bounds for them. Identifying a gauge with a convex body containing the origin in its interior (the unit ball of the gauge), we also prove that all introduced asymmetry measures are continuous in the Hausdorff distance. Finally, we show that, modifying one of the constructed asymmetry measures, a certain duality principle holds.