Bounding minimal solid angles of polytopes

TL;DR

This paper investigates the minimal solid angles of simplices in various dimensions, establishing bounds related to regular simplices and extending the analysis to polyhedral angles.

Contribution

It provides new bounds on minimal solid angles of simplices in dimensions three and four, and compares polyhedral angles to those of regular solids.

Findings

In dimension three, the minimal solid angle does not exceed that of the regular simplex.

In dimension four, the same bound holds for simplices close to the regular one.

The study extends to trihedral and dihedral angles of polyhedra.

Abstract

In this article we study the following question: What can be the measure of the minimal solid angle of a simplex in $\mathbb{R}^d$? We show that in dimensions three it is not greater than the solid angle of the regular simplex. And in dimension four the same holds for simplices sufficiently close to the regular simplex. We also study a similar question for trihedral and dihedral angles of polyhedra compared to those of regular solids.