On the circumradius of a special class of n-simplices

TL;DR

This paper derives a formula for the circumradius of circumscriptible n-simplices and establishes inequalities relating it to the edge-inradius, addressing a problem posed by the authors.

Contribution

It provides a closed-form expression for the circumradius and proves new inequalities, advancing understanding of the geometric properties of circumscriptible n-simplices.

Findings

Derived a closed formula for the circumradius.

Proved a double inequality involving circumradius and edge-inradius.

Confirmed part of a previously posed problem.

Abstract

An n-simplex is called circumscriptible (or edge-incentric) if there is a sphere tangent to all its n(n + 1)/2 edges. We obtain a closed formula for the radius of the circumscribed sphere of the circumscriptible n-simplex, and also prove a double inequality involving the circumradius and the edge-inradius of such simplices. Among this inequality settles affirmatively a part of a problem posed by the authors.