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

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.