Minkowski concentricity and complete simplices

TL;DR

This paper explores the relationships between radii functionals, Minkowski asymmetry, and simplices in Minkowski spaces, extending classical inequalities to non-symmetric gauge bodies and revealing new geometric connections.

Contribution

It generalizes the concentricity inequality for non-symmetric gauge bodies and links it to extremal sets and the completeness of simplices in Minkowski spaces.

Findings

Derived a generalized concentricity inequality for non-symmetric gauge bodies.

Established a connection between this inequality and extremal sets of the Bohnenblust inequality.

Revealed the relationship between the inequality and the completeness of simplices.

Abstract

This paper considers the radii functionals (circumradius, inradius, and diameter) as well as the Minkowski asymmetry for general (possibly non-symmetric) gauge bodies. A generalization of the concentricity inequality (which states that the sum of the inradius and circumradius is not greater than the diameter in general Minkowski spaces) for non-symmetric gauge bodies is derived and a strong connection between this new inequality, extremal sets of the generalized Bohnenblust inequality, and completeness of simplices is revealed.