Weakly Inscribed Polyhedra

TL;DR

This paper investigates convex polyhedra with vertices on a sphere in projective space, characterizing their combinatorial structure, dihedral angles, and induced hyperbolic-de Sitter geometry, expanding understanding of weakly inscribed polyhedra.

Contribution

It provides a combinatorial characterization of weakly inscribed polyhedra, describes their dihedral angles via linear programming, and details the hyperbolic-de Sitter structure on their boundary.

Findings

Characterization of 1-skeleta of weakly inscribed polyhedra

Linear programming description of dihedral angles

Description of hyperbolic-de Sitter boundary structure

Abstract

We study convex polyhedra in $\mathbb{R}\mathbb{P}^3$ with all their vertices on a sphere. We do not require, in particular, that the polyhedra lie in the interior of the sphere, hence the term "weakly inscribed". Such polyhedra can be interpreted as ideal polyhedra, if we regard $\mathbb{R}\mathbb{P}^3$ as a combination of the hyperbolic space and the de Sitter space, with the sphere as the common ideal boundary. We have three main results: (1) the $1$-skeleta of weakly inscribed polyhedra are characterized in a purely combinatorial way, (2) the exterior dihedral angles are characterized by linear programming, and (3) we also describe the hyperbolic-de Sitter structure induced on the boundary of weakly inscribed polyhedra.