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Polyhedra inscribed in a quadric ; ; Schlenker, Jean-Marc in Inventiones Mathematicae (2020), 221(1), 237-300 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 ... [more ▼] 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. 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. [less ▲] Detailed reference viewed: 81 (5 UL)Quasicircles and width of Jordan curves in CP1 ; ; et al E-print/Working paper (2019) We study a notion of "width" for Jordan curves in CP1, paying special attention to the class of quasicircles. The width of a Jordan curve is defined in terms of the geometry of its convex hull in ... [more ▼] We study a notion of "width" for Jordan curves in CP1, paying special attention to the class of quasicircles. The width of a Jordan curve is defined in terms of the geometry of its convex hull in hyperbolic three-space. A similar invariant in the setting of anti de Sitter geometry was used by Bonsante-Schlenker to characterize quasicircles amongst a larger class of Jordan curves in the boundary of anti de Sitter space. By contrast to the AdS setting, we show that there are Jordan curves of bounded width which fail to be quasicircles. However, we show that Jordan curves with small width are quasicircles. [less ▲] Detailed reference viewed: 31 (0 UL)The induced metric on the boundary of the convex hull of a quasicircle in hyperbolic and anti de Sitter geometry ; ; et al E-print/Working paper (2019) Celebrated work of Alexandrov and Pogorelov determines exactly which metrics on the sphere are induced on the boundary of a compact convex subset of hyperbolic three-space. As a step toward a ... [more ▼] Celebrated work of Alexandrov and Pogorelov determines exactly which metrics on the sphere are induced on the boundary of a compact convex subset of hyperbolic three-space. As a step toward a generalization for unbounded convex subsets, we consider convex regions of hyperbolic three-space bounded by two properly embedded disks which meet at infinity along a Jordan curve in the ideal boundary. In this setting, it is natural to augment the notion of induced metric on the boundary of the convex set to include a gluing map at infinity which records how the asymptotic geometry of the two surfaces compares near points of the limiting Jordan curve. Restricting further to the case in which the induced metrics on the two bounding surfaces have constant curvature Kâˆˆ[âˆ’1,0) and the Jordan curve at infinity is a quasicircle, the gluing map is naturally a quasisymmetric homeomorphism of the circle. The main result is that for each value of K, every quasisymmetric map is achieved as the gluing map at infinity along some quasicircle. We also prove analogous results in the setting of three-dimensional anti de Sitter geometry. Our results may be viewed as universal versions of the conjectures of Thurston and Mess about prescribing the induced metric on the boundary of the convex core of quasifuchsian hyperbolic manifolds and globally hyperbolic anti de Sitter spacetimes. [less ▲] Detailed reference viewed: 42 (2 UL)Higher signature Delaunay decompositions ; ; Schlenker, Jean-Marc E-print/Working paper (2016) A Delaunay decomposition is a cell decomposition in R^d for which each cell is inscribed in a Euclidean ball which is empty of all other vertices. This article introduces a generalization of the Delaunay ... [more ▼] A Delaunay decomposition is a cell decomposition in R^d for which each cell is inscribed in a Euclidean ball which is empty of all other vertices. This article introduces a generalization of the Delaunay decomposition in which the Euclidean balls in the empty ball condition are replaced by other families of regions bounded by certain quadratic hypersurfaces. This generalized notion is adaptable to geometric contexts in which the natural space from which the point set is sampled is not Euclidean, but rather some other flat semi-Riemannian geometry, possibly with degenerate directions. We prove the existence and uniqueness of the decomposition and discuss some of its basic properties. In the case of dimension d = 2, we study the extent to which some of the well-known optimality properties of the Euclidean Delaunay triangulation generalize to the higher signature setting. In particular, we describe a higher signature generalization of a well-known description of Delaunay decompositions in terms of the intersection angles between the circumscribed circles. [less ▲] Detailed reference viewed: 35 (2 UL) |
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