Extremal Radii, Diameter, and Minimum Width in Generalized Minkowski Spaces

TL;DR

This paper explores fundamental size measures like circumradius, inradius, diameter, and width in generalized Minkowski spaces, providing a theoretical framework for understanding geometric properties relative to gauges.

Contribution

It introduces a new theoretical approach to measure convex set sizes in generalized Minkowski spaces using containment and homothety concepts.

Findings

Defines size measures in generalized Minkowski spaces

Formulates containment problems involving homothetic bodies

Establishes a theoretical basis for metric problems in these spaces

Abstract

We discuss the notions of circumradius, inradius, diameter, and minimum width in generalized Minkowski spaces (that is, with respect to gauges), i.e., we measure the "size" of a given convex set in a finite-dimensional real vector space with respect to another convex set. This is done via formulating some kind of containment problem incorporating homothetic bodies of the latter set or strips bounded by parallel supporting hyperplanes thereof. The paper can be seen as a theoretical starting point for studying metrical problems of sets in generalized Minkowski spaces.