Tight bounds on angle sums of nonobtuse simplices

Abstract

It is widely known that the sum of the angles of a triangle equals two right angles. Far less known are the answers to similar questions for tetrahedra and higher dimensional simplices. In this paper we review some of these less known results, and look at them from a different point of view. Then we continue to derive tight bounds on the dihedral angle sums for the subclass of nonobtuse simplices. All the dihedral angles of such simplices are less than or equal to right. They have several important applications (Brandts et al., 2009). The main conclusion is that when the spatial dimension n is even, the range of dihedral angle sums of nonobtuse simplices is n times smaller than the corresponding range for arbitrary simplices. When n is odd, it is n-1 times smaller.