A complete system of inequalities for the diameter, in-, and circumradius in the 3-dimensional Euclidean space

2025/09/05 by Brandenberg, René, Merino, Bernardo González, Runge, Mia
#FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2509.05028

Abstract

We present a complete system of inequalities for the inradius, circumradius, and diameter in the 3-dimensional Euclidean space. To do so, we prove quasiconcavity of the inradius evaluated over n-simplices with a common facet independently of the norm/gauge under consideration.

Citations

• Complete systems of inequalities relating the perimeter, the area and the Cheeger constant of planar domains
• The missing (A, D, r) diagram
• Minkowski concentricity and complete simplices
• A complete 3-dimensional Blaschke-Santaló diagram

Related

• Minkowski concentricity and complete simplices
• Jung-type Inequalities and Blaschke-Santaló Diagrams for Different Diameter Variants
• Sharp inequalities involving the Cheeger constant of planar convex sets
• A remark on perimeter-diameter and perimeter-circumradius inequalities\n under lattice constraints
• Some inequalities for tetrahedra