Jung-type Inequalities and Blaschke-Santal\'o Diagrams for Different Diameter Variants

TL;DR

This paper investigates geometric inequalities involving circumradius and diameter for convex bodies under various gauges, exploring how different diameter definitions and symmetrizations affect these inequalities.

Contribution

It introduces new inequalities for convex bodies with respect to general gauges and analyzes the impact of different diameter definitions and symmetrizations.

Findings

Identifies multiple diameter definitions and their properties.

Establishes inequalities relating circumradius, diameter, and inradius.

Highlights differences in inequalities for symmetric and non-symmetric gauges.

Abstract

We study geometric inequalities for the circumradius and diameter with respect to general gauges, partly also involving the inradius and the Minkowski asymmetry. There are a number of options for defining the diameter of a convex body that fall apart when we consider non-symmetric gauges. These definitions correspond to different symmetrizations of the gauge, i.e. means of the gauge $C$ and its origin reflection $-C$.