Abstract

For an $n$-simplex, let $\alpha,\,\beta$ denote the maximum, and the minimum dihedral angles of the simplex, respectively. It is proved that the inequality $\alpha\le \arccos(1/n)\le \beta$ always holds, and either side equality implies that the $n$-simplex is a regular simplex. Similar inequalities are also given for a star-simplex, which is defined as a simplex that has a vertex (apex) such that the angles between distinct edges incident to the apex are all equal. Further, an explicit formula for the dihedral angle of a star-simplex between two distinct facets sharing the apex in common is presented in terms of the angle between two edges incident to the apex.

This work is licensed under a Creative Commons Attribution 4.0 License.