The ratios of diameter and width of reduced and of complete convex bodies in Minkowski spaces

Original Paper
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Abstract

The breadth of a convex body K within a normed space \({\mathbb {R}}^d\) in direction of a given hyperplane H is the distance between the two supporting hyperplanes of K parallel to H, measured in the underlying norm. The minimal and the maximal value over all directions are called minimal width and diameter of K. The body K is reduced if every body \(K_*\subsetneq K\) has a smaller minimal width, and K is complete if every body \(K^*\supsetneq K\) has a larger diameter. We study the ratio between diameter and minimal width of reduced and of complete bodies. If \(d \ge 3\) then there exist reduced bodies of arbitrarily large ratio, whereas the ratio for complete bodies is bounded by \(\frac{d+1}{2}\). As a consequence, every normed space \({\mathbb {R}}^d\), \(d \ge 2\), contains reduced bodies that are not complete.

Keywords

Minkowski space Convex body Reduced Complete Minimal width Diameter 

Mathematics Subject Classification

52A20 52A21 52A40 

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Copyright information

© The Managing Editors 2017

Authors and Affiliations

  1. 1.Institut für MathematikFriedrich-Schiller-Universität JenaJenaGermany

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