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

René Brandenberg; Bernardo González Merino; Mia Runge

paper

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

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.