On Characterizations of Some General $(\alpha, \beta)$ Norms in a Minkowski Space

TL;DR

This paper characterizes certain classes of Minkowski norms, specifically general $(eta, eta)$ norms, using geometric properties like Darboux curves and isoperimetric inequalities, with implications for Randers norms.

Contribution

It provides a new geometric characterization of 3D general $(eta, eta)$ norms and extends to global properties of Randers norms in higher dimensions.

Findings

Characterization of 3D general $(eta, eta)$ norms via Darboux curves.

Identification of global geometric quantities for Randers norms.

Use of isoperimetric inequalities to distinguish Minkowski norms.

Abstract

General $(\alpha, \beta)$ norms are an important class of Minkowski norms which contains the original $(\alpha, \beta)$ norms. In this note, by studying the behavior of the Darboux curves of the indicatrix, we give a characterization of 3-dimensional general $(\alpha, \beta)$ norms. By studying the isoperimetric properties of the indicatrix, as well as the isoperimetric inequalities in a Minkowski space, we give some global geometric quantities which characterizes Randers norms of arbitrary dimensions.