Isoperimetric Inequalities in Normed Planes

TL;DR

This paper extends the classical isoperimetric inequality to normed planes with smooth unit balls, providing improved inequalities for symmetric and constant width curves and analyzing the defect in terms of singular set areas.

Contribution

It introduces new isoperimetric inequalities in normed planes with smooth unit balls, focusing on symmetric and constant width curves and their singular sets.

Findings

Derived improved isoperimetric inequalities for specific curve classes

Identified conditions under which equality holds in these inequalities

Analyzed the defect in the inequality via singular set areas

Abstract

The classical isoperimetric inequality can be extended to a general normed plane. In the Euclidean plane, the defect in the isoperimetric inequality can be calculated in terms of the signed areas of some singular sets. In this paper we consider normed planes with smooth by parts unit balls and the corresponding class of admissible curves. For such an admissible curve, the singular sets are defined as projections in the subspaces of symmetric and constant width admissible curves. In this context, we obtain some improved isoperimetric inequalities whose equality hold for symmetric or constant width curves.