411 research outputs found

    On the number of tetrahedra with minimum, unit, and distinct volumes in three-space

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    We formulate and give partial answers to several combinatorial problems on volumes of simplices determined by nn points in 3-space, and in general in dd dimensions. (i) The number of tetrahedra of minimum (nonzero) volume spanned by nn points in \RR^3 is at most 2/3n3O(n2){2/3}n^3-O(n^2), and there are point sets for which this number is 3/16n3O(n2){3/16}n^3-O(n^2). We also present an O(n3)O(n^3) time algorithm for reporting all tetrahedra of minimum nonzero volume, and thereby extend an algorithm of Edelsbrunner, O'Rourke, and Seidel. In general, for every k,d\in \NN, 1kd1\leq k \leq d, the maximum number of kk-dimensional simplices of minimum (nonzero) volume spanned by nn points in \RR^d is Θ(nk)\Theta(n^k). (ii) The number of unit-volume tetrahedra determined by nn points in \RR^3 is O(n7/2)O(n^{7/2}), and there are point sets for which this number is Ω(n3loglogn)\Omega(n^3 \log \log{n}). (iii) For every d\in \NN, the minimum number of distinct volumes of all full-dimensional simplices determined by nn points in \RR^d, not all on a hyperplane, is Θ(n)\Theta(n).Comment: 19 pages, 3 figures, a preliminary version has appeard in proceedings of the ACM-SIAM Symposium on Discrete Algorithms, 200

    Maximal Area Triangles in a Convex Polygon

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    The widely known linear time algorithm for computing the maximum area triangle in a convex polygon was found incorrect recently by Keikha et. al.(arXiv:1705.11035). We present an alternative algorithm in this paper. Comparing to the only previously known correct solution, ours is much simpler and more efficient. More importantly, our new approach is powerful in solving related problems

    Triangle areas in line arrangements

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    A widely investigated subject in combinatorial geometry, originated from Erd\H{o}s, is the following. Given a point set PP of cardinality nn in the plane, how can we describe the distribution of the determined distances? This has been generalized in many directions. In this paper we propose the following variants. Consider planar arrangements of nn lines. Determine the maximum number of triangles of unit area, maximum area or minimum area, determined by these lines. Determine the minimum size of a subset of these nn lines so that all triples determine distinct area triangles. We prove that the order of magnitude for the maximum occurrence of unit areas lies between Ω(n2)\Omega(n^2) and O(n9/4)O(n^{9/4}). This result is strongly connected to both additive combinatorial results and Szemer\'edi--Trotter type incidence theorems. Next we show a tight bound for the maximum number of minimum area triangles. Finally we present lower and upper bounds for the maximum area and distinct area problems by combining algebraic, geometric and combinatorial techniques.Comment: Title is shortened. Some typos and small errors were correcte

    Density estimates of 1-avoiding sets via higher order correlations

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    We improve the best known upper bound on the density of a planar measurable set A containing no two points at unit distance to 0.25442. We use a combination of Fourier analytic and linear programming methods to obtain the result. The estimate is achieved by means of obtaining new linear constraints on the autocorrelation function of A utilizing triple-order correlations in A, a concept that has not been previously studied.Comment: 10 pages, 2 figure

    Rectangular Layouts and Contact Graphs

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    Contact graphs of isothetic rectangles unify many concepts from applications including VLSI and architectural design, computational geometry, and GIS. Minimizing the area of their corresponding {\em rectangular layouts} is a key problem. We study the area-optimization problem and show that it is NP-hard to find a minimum-area rectangular layout of a given contact graph. We present O(n)-time algorithms that construct O(n2)O(n^2)-area rectangular layouts for general contact graphs and O(nlogn)O(n\log n)-area rectangular layouts for trees. (For trees, this is an O(logn)O(\log n)-approximation algorithm.) We also present an infinite family of graphs (rsp., trees) that require Ω(n2)\Omega(n^2) (rsp., Ω(nlogn)\Omega(n\log n)) area. We derive these results by presenting a new characterization of graphs that admit rectangular layouts using the related concept of {\em rectangular duals}. A corollary to our results relates the class of graphs that admit rectangular layouts to {\em rectangle of influence drawings}.Comment: 28 pages, 13 figures, 55 references, 1 appendi

    Introducing symplectic billiards

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    In this article we introduce a simple dynamical system called symplectic billiards. As opposed to usual/Birkhoff billiards, where length is the generating function, for symplectic billiards symplectic area is the generating function. We explore basic properties and exhibit several similarities, but also differences of symplectic billiards to Birkhoff billiards.Comment: 41 pages, 16 figure

    Infinite N phase transitions in continuum Wilson loop operators

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    We define smoothed Wilson loop operators on a four dimensional lattice and check numerically that they have a finite and nontrivial continuum limit. The continuum operators maintain their character as unitary matrices and undergo a phase transition at infinite N reflected by the eigenvalue distribution closing a gap in its spectrum when the defining smooth loop is dilated from a small size to a large one. If this large N phase transition belongs to a solvable universality class one might be able to calculate analytically the string tension in terms of the perturbative Lambda-parameter. This would be achieved by matching instanton results for small loops to the relevant large-N-universal function which, in turn, would be matched for large loops to an effective string theory. Similarities between our findings and known analytical results in two dimensional space-time indicate that the phase transitions we found only affect the eigenvalue distribution, but the traces of finite powers of the Wilson loop operators stay smooth under scaling.Comment: 31 pages, 9 figures, typos and references corrected, minor clarifications adde

    A glimpse of the conformal structure of random planar maps

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    We present a way to study the conformal structure of random planar maps. The main idea is to explore the map along an SLE (Schramm--Loewner evolution) process of parameter κ=6 \kappa = 6 and to combine the locality property of the SLE_{6} together with the spatial Markov property of the underlying lattice in order to get a non-trivial geometric information. We follow this path in the case of the conformal structure of random triangulations with a boundary. Under a reasonable assumption called (*) that we have unfortunately not been able to verify, we prove that the limit of uniformized random planar triangulations has a fractal boundary measure of Hausdorff dimension 13\frac{1}{3} almost surely. This agrees with the physics KPZ predictions and represents a first step towards a rigorous understanding of the links between random planar maps and the Gaussian free field (GFF).Comment: To appear in Commun. Math. Phy

    Symmetry-break in Voronoi tessellations

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    We analyse in a common framework the properties of the Voronoi tessellations resulting from regular 2D and 3D crystals and those of tessellations generated by Poisson distributions of points, thus joining on symmetry breaking processes and the approach to uniform random distributions of seeds. We perturb crystalline structures in 2D and 3D with a spatial Gaussian noise whose adimensional strength is α and analyse the statistical properties of the cells of the resulting Voronoi tessellations using an ensemble approach. In 2D we consider triangular, square and hexagonal regular lattices, resulting into hexagonal, square and triangular tessellations, respectively. In 3D we consider the simple cubic (SC), body-centred cubic (BCC), and face-centred cubic (FCC) crystals, whose corresponding Voronoi cells are the cube, the truncated octahedron, and the rhombic dodecahedron, respectively. In 2D, for all values α>0, hexagons constitute the most common class of cells. Noise destroys the triangular and square tessellations, which are structurally unstable, as their topological properties are discontinuous in α=0. On the contrary, the honeycomb hexagonal tessellation is topologically stable and, experimentally, all Voronoi cells are hexagonal for small but finite noise with α0.5), memory of the specific initial unperturbed state is lost, because the statistical properties of the three perturbed regular tessellations are indistinguishable. When α>2, results converge to those of Poisson-Voronoi tessellations. In 2D, while the isoperimetric ratio increases with noise for the perturbed hexagonal tessellation, for the perturbed triangular and square tessellations it is optimised for specific value of noise intensity. The same applies in 3D, where noise degrades the isoperimetric ratio for perturbed FCC and BCC lattices, whereas the opposite holds for perturbed SCC lattices. This allows for formulating a weaker form of the Kelvin conjecture. By analysing jointly the statistical properties of the area and of the volume of the cells, we discover that also the cells shape heavily fluctuates when noise is introduced in the system. In 2D, the geometrical properties of n-sided cells change with α until the Poisson-Voronoi limit is reached for α>2; in this limit the Desch law for perimeters is shown to be not valid and a square root dependence on n is established, which agrees with exact asymptotic results. Anomalous scaling relations are observed between the perimeter and the area in the 2D and between the areas and the volumes of the cells in 3D: except for the hexagonal (2D) and FCC structure (3D), this applies also for infinitesimal noise. In the Poisson-Voronoi limit, the anomalous exponent is about 0.17 in both the 2D and 3D case. A positive anomaly in the scaling indicates that large cells preferentially feature large isoperimetric quotients. As the number of faces is strongly correlated with the sphericity (cells with more faces are bulkier), in 3D it is shown that the anomalous scaling is heavily reduced when we perform power law fits separately on cells with a specific number of faces
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