A Pseudopolynomial Algorithm for Alexandrov's Theorem

Alexandrov's Theorem states that every metric with the global topology and local geometry required of a convex polyhedron is in fact the intrinsic metric of a unique convex polyhedron. Recent work by Bobenko and Izmestiev describes a differential equation whose solution leads to the polyhedron corresponding to a given metric. We describe an algorithm based on this differential equation to compute the polyhedron to arbitrary precision given the metric, and prove a pseudopolynomial bound on its running time. Along the way, we develop pseudopolynomial algorithms for computing shortest paths and weighted Delaunay triangulations on a polyhedral surface, even when the surface edges are not shortest paths.Comment: 25 pages; new Delaunay triangulation algorithm, minor other changes; an abbreviated v2 was at WADS 200

Hall's Conjecture on Extremal Sets for Random Triangles

In this paper we partially resolve Hall's conjecture about the distribution of random triangles. We consider the probability that three points chosen uniformly at random, in a bounded convex region of the plane, form an acute triangle. Hall's conjecture is the "isoprobabilistic inequality" which states that this probability should be maximized by the disk. We first prove that the disk is a weak local maximum for planar regions and that the ball is a weak local maximum in three dimensional regions. We then prove a local $C^{2,1/2}$-type estimate on the probability in the Hausdorff topology. This enables us to prove that the disk is a local maximum in the Gromov-Hausdorff topology (modulo congruences). Finally, we give an explicit upper bound on the isoperimetric ratio of the regions which maximize the probability and show how this reduces verifying the full conjecture to a finite, though currently intractable, calculation. An interesting aspect of our work is the use of tools from outside of geometric probability. We use an autocorrelation integral to provide the appropriate framework for the problem. When we study the problem in $\mathbb{R}^3$, we need non-Abelian harmonic analysis. To set up the proof that the disk is a strong local maximum, we use the Borsak-Ulam theorem.Comment: In this version, we have reorganized the paper to have the technical proofs in the appendix. We have also included links to Mathematica code used to perform some of the calculation

Asymptotics of generalized Hadwiger numbers

We give asymptotic estimates for the number of non-overlapping homothetic copies of some centrally symmetric oval $B$ which have a common point with a 2-dimensional domain $F$ having rectifiable boundary, extending previous work of the L.Fejes-Toth, K.Borockzy Jr., D.G.Larman, S.Sezgin, C.Zong and the authors. The asymptotics compute the length of the boundary $\partial F$ in the Minkowski metric determined by $B$. The core of the proof consists of a method for sliding convex beads along curves with positive reach in the Minkowski plane. We also prove that level sets are rectifiable subsets, extending a theorem of Erd\"os, Oleksiv and Pesin for the Euclidean space to the Minkowski space.Comment: 20p, 9 figure

Covering a closed curve with a given total curvature

http://archive.org/details/coveringclosedcu00ghanN