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When I first met Oded Schramm in January 1991 at the University of California, San Diego, he introduced himself as a “Circle Packer”. This modest description referred to his Ph.D. thesis around the Koebe-Andreev-Thurston theorem and a discrete version of the Riemann mapping theorem, explained below. In a series of highly original papers, some joint with Zhen-Xu He, he created powerful new tools out...
The problem of illuminating the boundary of sets having constant width is considered and a bound for the number of directions needed is given. As a corollary, an estimate for Borsuk’s partition problem is inferred. Also, the illumination number of sufficiently symmetric strictly convex bodies is determined.
The nerve of a packing is a graph that encodes its combinatorics. The vertices of the nerve correspond to the packed sets, and an edge occurs between two vertices in the nerve precisely when the corresponding sets of the packing intersect. The nerve of a circle packing and other well-behaved packings, on the sphere or in the plane, is a planar graph. It was an observation of Thurston [Th1,...
Summary This paper proves that given a convex polyhedron $$P \subset \mathbb{R}^3 $$ and a smooth strictly convex body $$K \subset \mathbb{R}^3 $$ , there is some convex polyhedron Q combinatorically equivalent to P which midscribes K; that is, all the edges of Q are tangent to K. Furthermore, with some stronger smoothness conditions on ∂K, the space of all such Q is a six dimensional differentiable...
A domain in the Riemann sphere $$\hat{\mathbb{C}}$$ is called a circle domain if every connected component of its boundary is either a circle or a point. In 1908, P. Koebe [Ko1] posed the following conjecture, known as Koebe’s Kreisnormierungsproblem: Any plane domain is conformally homeomorphic to a circle domain in $$\hat{\mathbb{C}}$$ . When the domain is simply connected, this is the...
The contacts graph, or nerve, of a packing, is a combinatorial graph that describes the combinatorics of the packing. Let G be the 1-skeleton of a triangulation of an open disk. G is said to be CP parabolic (resp. CP hyperbolic) if there is a locally finite disk packing P in the plane (resp. the unit disk) with contacts graph G. Several criteria for deciding whether G is CP parabolic or CP hyperbolic...
Let $$\Omega \mathop \subset \limits_ \ne C$$ be a simply-connected domain. The Rodin–Sullivan Theorem states that a sequence of disk packings in the unit disk U converges, in a well-defined sense, to a conformal map from Ω to U. Moreover, it is known that the first and second derivatives converge as well. Here, it is proven that for hexagonal disk packings the convergence is C∞. This...
It is shown that a Gromov hyperbolic geodesic metric space X with bounded growth at some scale is roughly quasi-isometric to a convex subset of hyperbolic space. If one is allowed to rescale the metric of X by some positive constant, then there is an embedding where distances are distorted by at most an additive constant. Another embedding theorem states that any δ-hyperbolic metric space embeds...
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