Behaviour of inradius, circumradius, and diameter in generalized Minkowski spaces

Abstract

We provide a study of Blaschke–Santaló diagrams, i.e. complete systems of inequalities, for the inradius, diameter, and circumradius, measured with respect to different gauges. This contrasts previous works on those diagrams, which are all considered for the Euclidean measure. By proving several new inequalities and properties between these three functionals, we compute the intersection and the union over all possible gauges of those diagrams, showing that they coincide with the corresponding diagrams of a parallelotope and (in the planar case) a triangle, respectively. We also show that the planar spaces whose unit balls are regular pentagons or hexagons play an important role in the understanding of further extreme behaviours.