Weakly inscribed polyhedra

Chen, Hao; Schlenker, Jean-Marc

doi:10.1090/btran/59

Weakly inscribed polyhedra
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by Hao Chen and Jean-Marc Schlenker HTML | PDF

Trans. Amer. Math. Soc. Ser. B 9 (2022), 415-449

Abstract:

Motivated by an old question of Steiner, we study convex polyhedra in $\mathbb {R}\mathrm {P}^3$ with all their vertices on a sphere, but the polyhedra themselves do not lie on one side the sphere. We give an explicit combinatorial description of the possible combinatorics of such polyhedra. The proof uses a natural extension of the 3-dimensional hyperbolic space by the de Sitter space. Polyhedra with their vertices on the sphere are interpreted as ideal polyhedra in this extended space. We characterize the possible dihedral angles of those ideal polyhedra, as well as the geometric structures induced on their boundaries, which is composed of hyperbolic and de Sitter regions glued along their ideal boundaries.

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