Weakly inscribed polyhedra

Abstract

Motivated by an old question of Steiner, we study convex polyhedra in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R normal upper P cubed"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">P</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {R}\mathrm {P}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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.

Locations

• Transactions of the American Mathematical Society Series B - View - PDF
• arXiv (Cornell University) - View - PDF