5,250 research outputs found

    Finite convex geometries of circles

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    Let F be a finite set of circles in the plane. We point out that the usual convex closure restricted to F yields a convex geometry, that is, a combinatorial structure introduced by P. H Edelman in 1980 under the name "anti-exchange closure system". We prove that if the circles are collinear and they are arranged in a "concave way", then they determine a convex geometry of convex dimension at most 2, and each finite convex geometry of convex dimension at most 2 can be represented this way. The proof uses some recent results from Lattice Theory, and some of the auxiliary statements on lattices or convex geometries could be of separate interest. The paper is concluded with some open problems.Comment: 22 pages, 7 figure

    A better upper bound on the number of triangulations of a planar point set

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    We show that a point set of cardinality nn in the plane cannot be the vertex set of more than 59nO(n6)59^n O(n^{-6}) straight-edge triangulations of its convex hull. This improves the previous upper bound of 276.75n276.75^n.Comment: 6 pages, 1 figur

    The polytope of non-crossing graphs on a planar point set

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    For any finite set \A of nn points in R2\R^2, we define a (3n3)(3n-3)-dimensional simple polyhedron whose face poset is isomorphic to the poset of ``non-crossing marked graphs'' with vertex set \A, where a marked graph is defined as a geometric graph together with a subset of its vertices. The poset of non-crossing graphs on \A appears as the complement of the star of a face in that polyhedron. The polyhedron has a unique maximal bounded face, of dimension 2ni+n32n_i +n -3 where nin_i is the number of points of \A in the interior of \conv(\A). The vertices of this polytope are all the pseudo-triangulations of \A, and the edges are flips of two types: the traditional diagonal flips (in pseudo-triangulations) and the removal or insertion of a single edge. As a by-product of our construction we prove that all pseudo-triangulations are infinitesimally rigid graphs.Comment: 28 pages, 16 figures. Main change from v1 and v2: Introduction has been reshape

    Representing convex geometries by almost-circles

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    Finite convex geometries are combinatorial structures. It follows from a recent result of M.\ Richter and L.G.\ Rogers that there is an infinite set TrrT_{rr} of planar convex polygons such that TrrT_{rr} with respect to geometric convex hulls is a locally convex geometry and every finite convex geometry can be represented by restricting the structure of TrrT_{rr} to a finite subset in a natural way. An \emph{almost-circle of accuracy} 1ϵ1-\epsilon is a differentiable convex simple closed curve SS in the plane having an inscribed circle of radius r1>0r_1>0 and a circumscribed circle of radius r2r_2 such that the ratio r1/r2r_1/r_2 is at least 1ϵ1-\epsilon. % Motivated by Richter and Rogers' result, we construct a set TnewT_{new} such that (1) TnewT_{new} contains all points of the plane as degenerate singleton circles and all of its non-singleton members are differentiable convex simple closed planar curves; (2) TnewT_{new} with respect to the geometric convex hull operator is a locally convex geometry; (3) as opposed to TrrT_{rr}, TnewT_{new} is closed with respect to non-degenerate affine transformations; and (4) for every (small) positive ϵ\epsilon\in\real and for every finite convex geometry, there are continuum many pairwise affine-disjoint finite subsets EE of TnewT_{new} such that each EE consists of almost-circles of accuracy 1ϵ1-\epsilon and the convex geometry in question is represented by restricting the convex hull operator to EE. The affine-disjointness of E1E_1 and E2E_2 means that, in addition to E1E2=E_1\cap E_2=\emptyset, even ψ(E1)\psi(E_1) is disjoint from E2E_2 for every non-degenerate affine transformation ψ\psi.Comment: 18 pages, 6 figure

    On the Complexity of Polytope Isomorphism Problems

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    We show that the problem to decide whether two (convex) polytopes, given by their vertex-facet incidences, are combinatorially isomorphic is graph isomorphism complete, even for simple or simplicial polytopes. On the other hand, we give a polynomial time algorithm for the combinatorial polytope isomorphism problem in bounded dimensions. Furthermore, we derive that the problems to decide whether two polytopes, given either by vertex or by facet descriptions, are projectively or affinely isomorphic are graph isomorphism hard. The original version of the paper (June 2001, 11 pages) had the title ``On the Complexity of Isomorphism Problems Related to Polytopes''. The main difference between the current and the former version is a new polynomial time algorithm for polytope isomorphism in bounded dimension that does not rely on Luks polynomial time algorithm for checking two graphs of bounded valence for isomorphism. Furthermore, the treatment of geometric isomorphism problems was extended.Comment: 16 pages; to appear in: Graphs and Comb.; replaces our paper ``On the Complexity of Isomorphism Problems Related to Polytopes'' (June 2001

    Bounding Helly numbers via Betti numbers

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    We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem. More precisely, we show that for any non-negative integers bb and dd there exists an integer h(b,d)h(b,d) such that the following holds. If F\mathcal F is a finite family of subsets of Rd\mathbb R^d such that β~i(G)b\tilde\beta_i\left(\bigcap\mathcal G\right) \le b for any GF\mathcal G \subsetneq \mathcal F and every 0id/210 \le i \le \lceil d/2 \rceil-1 then F\mathcal F has Helly number at most h(b,d)h(b,d). Here β~i\tilde\beta_i denotes the reduced Z2\mathbb Z_2-Betti numbers (with singular homology). These topological conditions are sharp: not controlling any of these d/2\lceil d/2 \rceil first Betti numbers allow for families with unbounded Helly number. Our proofs combine homological non-embeddability results with a Ramsey-based approach to build, given an arbitrary simplicial complex KK, some well-behaved chain map C(K)C(Rd)C_*(K) \to C_*(\mathbb R^d).Comment: 29 pages, 8 figure
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