On $(r,D,R)$-Blaschke-Santal\'o diagrams with regular $k$-gon gauges

TL;DR

This paper explores Blaschke-Santaló diagrams for inradius, diameter, and circumradius using various gauges, revealing new inequalities and extremal properties, especially for regular polygons, and characterizing their intersections and unions.

Contribution

It introduces novel inequalities and properties for these diagrams under different gauges, extending classical Euclidean results to regular polygon gauges and identifying extremal configurations.

Findings

Diagrams for regular polygons coincide with those of parallelotopes and triangles.

New inequalities relate inradius, diameter, and circumradius under various gauges.

Extremal properties are established for regular pentagon and hexagon gauges.

Abstract

We provide a study of Blaschke-Santal\'o diagrams for the inradius, diameter, and circumradius, measured with respect to different gauges. This contrasts previous works on those diagrams, which are all considered for 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. Further new extremal properties are derived considering the diagrams with respect to a regular pentagon or hexagon gauge.