Polyhedra inscribed in a quadric

• 成果类型: Article
• 署名作者: Danciger, Jeffrey; Maloni, Sara; Schlenker, Jean-Marc
• 署名单位: University of Texas System; University of Texas Austin; University of Virginia; University of Luxembourg
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-020-00948-9
• 发表日期: 2020
• 页码: 237-300
• 关键词: ideal polyhedra; spacetimes; collisions; particles; minima; MODULI; lines

摘要

We study convex polyhedra in three-space that are inscribed in a quadric surface. Up to projective transformations, there are three such surfaces: the sphere, the hyperboloid, and the cylinder. Our main result is that a planar graph Gamma is realized as the 1-skeleton of a polyhedron inscribed in the hyperboloid or cylinder if and only if Gamma is realized as the 1-skeleton of a polyhedron inscribed in the sphere and Gamma admits a Hamiltonian cycle. This answers a question asked by Steiner in 1832. Rivin characterized convex polyhedra inscribed in the sphere by studying the geometry of ideal polyhedra in hyperbolic space. We study the case of the hyperboloid and the cylinder by parameterizing the space of convex ideal polyhedra in anti-de Sitter geometry and in half-pipe geometry. Just as the cylinder can be seen as a degeneration of the sphere and the hyperboloid, half-pipe geometry is naturally a limit of both hyperbolic and anti-de Sitter geometry. We promote a unified point of view to the study of the three cases throughout.