Successive Radii and Ball Operators in Generalized Minkowski Spaces

TL;DR

This paper explores properties of successive radii and ball operators in generalized Minkowski spaces, focusing on minimal containment problems, and their connections to minimax location, diametrical completeness, and constant width.

Contribution

It introduces new insights into the behavior of successive radii and ball operators within generalized Minkowski spaces, linking geometric measures to location and completeness problems.

Findings

Characterization of successive radii in generalized Minkowski spaces

Analysis of ball intersections and ball hulls in this context

Connections established between radii, minimax location, and constant width

Abstract

We investigate elementary properties of successive radii 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 minimal containment problems, where intersections or Minkowski sums of the latter set and affine flats of a certain dimension are incorporated. Since this is strongly related to minimax location problems and to the notions of diametrical completeness and constant width, we also have a look at ball intersections and ball hulls.