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  • Equivelar Toroids with Few Flag-Orbits
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2021-01-25
    José Collins, Antonio Montero

    An \((n+1)\)-toroid is a quotient of a tessellation of the n-dimensional Euclidean space with a lattice group. Toroids are generalisations of maps on the torus to higher dimensions and also provide examples of abstract polytopes. Equivelar toroids are those that are induced by regular tessellations. In this paper we present a classification of equivelar \((n+1)\)-toroids with at most n flag-orbits;

    更新日期:2021-01-25
  • Large Equilateral Sets in Subspaces of $$\ell _\infty ^n$$ ℓ ∞ n of Small Codimension
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2021-01-22
    Nóra Frankl

    For fixed k we prove exponential lower bounds on the equilateral number of subspaces of \(\ell _{\infty }^n\) of codimension k. In particular, we show that subspaces of codimension 2 of \(\ell _{\infty }^{n+2}\) and subspaces of codimension 3 of \(\ell _{\infty }^{n+3}\) have an equilateral set of cardinality \(n+1\) if \(n\ge 7\) and \(n\ge 12\) respectively. Moreover, the same is true for every normed

    更新日期:2021-01-22
  • Lattice Size and Generalized Basis Reduction in Dimension Three
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2021-01-20
    Anthony Harrison, Jenya Soprunova

    The lattice size of a lattice polytope P was defined and studied by Schicho, and Castryck and Cools. They provided an “onion skins” algorithm for computing the lattice size of a lattice polygon P in \(\mathbb R^2\) based on passing successively to the convex hull of the interior lattice points of P. We explain the connection of the lattice size to the successive minima of \(K=(P+(-P))^*\) and to the

    更新日期:2021-01-20
  • Combinatorial Modifications of Reeb Graphs and the Realization Problem
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2021-01-13
    Łukasz Patryk Michalak

    We prove that, up to homeomorphism, any graph subject to natural necessary conditions on orientation and the cycle rank can be realized as the Reeb graph of a Morse function on a given closed manifold M. Along the way, we show that the Reeb number \(\mathcal {R}(M)\), i.e., the maximum cycle rank among all Reeb graphs of functions on M, is equal to the corank of fundamental group \(\pi _1(M)\), thus

    更新日期:2021-01-14
  • On the Falk Invariant of Shi and Linial Arrangements
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2021-01-08
    Weili Guo, Michele Torielli

    It is an open question to give a combinatorial interpretation of the Falk invariant of a hyperplane arrangement, i.e., the third rank of successive quotients in the lower central series of the fundamental group of the arrangement. In this article, we give a combinatorial formula for this invariant in the case of hyperplane arrangements that are complete lift representations of certain gain graphs.

    更新日期:2021-01-08
  • On Visibility Problems with an Infinite Discrete Set of Obstacles
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2021-01-05
    Michael Boshernitzan, Yaar Solomon

    This paper deals with visibility problems in Euclidean spaces where the set of obstacles Y is an infinite discrete point set. We prove five independent results. Consider the following problem. Given \(\varepsilon >0\), imagine a forest whose trees have radius \(\varepsilon \) and their locations are given by the set Y. Suppose that a light source is at infinity, and that there are no arbitrarily large

    更新日期:2021-01-05
  • On Arithmetic Progressions in Model Sets
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2021-01-04
    Anna Klick, Nicolae Strungaru, Adi Tcaciuc

    We establish the existence of arbitrary-length arithmetic progressions in model sets and Meyer sets in Euclidean d-space. We prove a van der Waerden-type theorem for Meyer sets. We show that subsets of Meyer sets with positive density and pure point diffraction contain arithmetic progressions of arbitrary length.

    更新日期:2021-01-05
  • On the Lattice Hadwiger Number of Superballs and Some Other Bodies
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2021-01-03
    Serge Vlăduţ

    We show that the lattice Hadwiger (= kissing) number of superballs is exponential in the dimension. The same methods can be used to show exponential growth for more general convex bodies as well.

    更新日期:2021-01-03
  • Octahedralizing 3-Colorable 3-Polytopes
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2021-01-03
    Giulia Codenotti, Lorenzo Venturello

    We investigate the question of whether any d-colorable simplicial d-polytope can be octahedralized, i.e., can be subdivided to a d-dimensional geometric cross-polytopal complex. We give a positive answer in dimension 3, with the additional property that the octahedralization introduces no new vertices on the boundary of the polytope.

    更新日期:2021-01-03
  • Fitting Tractable Convex Sets to Support Function Evaluations
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2021-01-03
    Yong Sheng Soh, Venkat Chandrasekaran

    The geometric problem of estimating an unknown compact convex set from evaluations of its support function arises in a range of scientific and engineering applications. Traditional approaches typically rely on estimators that minimize the error over all possible compact convex sets; in particular, these methods allow for limited incorporation of prior structural information about the underlying set

    更新日期:2021-01-03
  • Improvement on the Crossing Number of Crossing-Critical Graphs
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-12-18
    János Barát, Géza Tóth

    The crossing number of a graph G is the minimum number of edge crossings over all drawings of G in the plane. A graph G is k-crossing-critical if its crossing number is at least k, but if we remove any edge of G, its crossing number drops below k. There are examples of k-crossing-critical graphs that do not have drawings with exactly k crossings. Richter and Thomassen proved in 1993 that if G is k-crossing-critical

    更新日期:2020-12-18
  • Triangulating Submanifolds: An Elementary and Quantified Version of Whitney’s Method
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-12-11
    Jean-Daniel Boissonnat, Siargey Kachanovich, Mathijs Wintraecken

    We quantise Whitney’s construction to prove the existence of a triangulation for any \(C^2\) manifold, so that we get an algorithm with explicit bounds. We also give a new elementary proof, which is completely geometric.

    更新日期:2020-12-12
  • 更新日期:2020-12-12
  • Density Estimates of 1-Avoiding Sets via Higher Order Correlations
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-12-10
    Gergely Ambrus, Máté Matolcsi

    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

    更新日期:2020-12-10
  • Topologically Trivial Closed Walks in Directed Surface Graphs
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-11-30
    Jeff Erickson, Yipu Wang

    Let G be a directed graph with n vertices and m edges, embedded on a surface S, possibly with boundary, with first Betti number \(\beta \). We consider the complexity of finding closed directed walks in G that are either contractible (trivial in homotopy) or bounding (trivial in integer homology) in S. Specifically, we describe algorithms to determine whether G contains a simple contractible cycle

    更新日期:2020-12-01
  • A Sylvester–Gallai Result for Concurrent Lines in the Complex Plane
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-11-06
    Alex Cohen

    We show that if a finite non-collinear set of points in \(\mathbb {C}^2\) lies on a family of m concurrent lines, and if one of those lines contains more than \(m-2\) points, there exists a line passing through exactly two points of the set. The bound \(m-2\) in our result is optimal. Our main theorem resolves a conjecture of Frank de Zeeuw, and generalizes a result of Kelly and Nwankpa.

    更新日期:2020-11-06
  • Balanced Convex Partitions of Lines in the Plane
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-11-05
    Alexander Xue, Pablo Soberón

    We prove an extension of a ham sandwich theorem for families of lines in the plane by Dujmović and Langerman. Given two sets A, B of n lines each in the plane, we prove that it is possible to partition the plane into r closed convex regions so that the following holds. For each region C of the partition there is a subset of \(c_r n^{1/r}\) lines of A whose pairwise intersections are in C, and the same

    更新日期:2020-11-06
  • Asymptotical Unboundedness of the Heesch Number in $${\mathbb {E}}^d$$ E d for $$d\rightarrow \infty $$ d → ∞
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-11-02
    Bojan Bašić, Anna Slivková

    We solve d-dimensional Heesch’s problem in the asymptotic sense. Namely, we show that, if \(d\rightarrow \infty \), then there is no uniform upper bound on the set of all possible finite Heesch numbers in the space \({\mathbb {E}}^d\); in other words, given any nonnegative integer n, we can find a dimension d (depending on n) in which there exists a hypersolid whose Heesch number is finite and greater

    更新日期:2020-11-03
  • Tilings of Convex Polyhedral Cones and Topological Properties of Self-Affine Tiles
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-10-16
    Ya-min Yang, Yuan Zhang

    Let \({\varvec{a}}_1,\ldots , {\varvec{a}}_r\) be vectors in a half-space of \({\mathbb {R}}^n\). We call \(C={\varvec{a}}_1{\mathbb {R}}^{+}+\cdots +{\varvec{a}}_r {\mathbb {R}}^{+}\) a convex polyhedral cone and \(\{{\varvec{a}}_1,\ldots , {\varvec{a}}_r\}\) a generator set of C. A generator set with the minimal cardinality is called a frame. We investigate the translation tilings of convex polyhedral

    更新日期:2020-10-17
  • Counterexample to a Variant of a Conjecture of Ziegler Concerning a Simple Polytope and Its Dual
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-10-15
    William Gustafson

    Problem 4.19 in Ziegler (Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, New York (1995)) asserts that every simple 3-dimensional polytope has the property that its dual can be constructed as the convex hull of points chosen from the facets of the original polytope. In this note we state a variant of this conjecture that requires the points to be a subset of the vertices of

    更新日期:2020-10-16
  • Counting Polygon Triangulations is Hard
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-10-13
    David Eppstein

    We prove that it is \(\#{\mathsf {P}}\)-complete to count the triangulations of a (non-simple) polygon.

    更新日期:2020-10-14
  • Classification of Triples of Lattice Polytopes with a Given Mixed Volume
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-10-13
    Gennadiy Averkov, Christopher Borger, Ivan Soprunov

    We present an algorithm for the classification of triples of lattice polytopes with a given mixed volume m in dimension 3. It is known that the classification can be reduced to the enumeration of so-called irreducible triples, the number of which is finite for fixed m. Following this algorithm, we enumerate all irreducible triples of normalized mixed volume up to 4 that are inclusion-maximal. This

    更新日期:2020-10-13
  • Randomized Construction of Complexes with Large Diameter
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-09-23
    Francisco Criado, Andrew Newman

    We consider the question of the largest possible combinatorial diameter among pure dimensional and strongly connected \((d-1)\)-dimensional simplicial complexes on n vertices, denoted \(H_s(n, d)\). Using a probabilistic construction we give a new lower bound on \(H_s(n, d)\) that is within an \(O(d^2)\) factor of the upper bound. This improves on the previously best known lower bound which was within

    更新日期:2020-09-23
  • Dynamic Planar Voronoi Diagrams for General Distance Functions and Their Algorithmic Applications
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-09-22
    Haim Kaplan, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, Micha Sharir

    We describe a new data structure for dynamic nearest neighbor queries in the plane with respect to a general family of distance functions. These include \(L_p\)-norms and additively weighted Euclidean distances. Our data structure supports general (convex, pairwise disjoint) sites that have constant description complexity (e.g., points, line segments, disks, etc.). Our structure uses \(O(n \log ^3

    更新日期:2020-09-23
  • Packing Disks by Flipping and Flowing
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-09-14
    Robert Connelly, Steven J. Gortler

    We provide a new type of proof for the Koebe–Andreev–Thurston (KAT) planar circle packing theorem based on combinatorial edge-flips. In particular, we show that starting from a disk packing with a maximal planar contact graph G, one can remove any flippable edge \(e^-\) of this graph and then continuously flow the disks in the plane, so that at the end of the flow, one obtains a new disk packing whose

    更新日期:2020-09-14
  • Randomized Incremental Construction of Delaunay Triangulations of Nice Point Sets
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-09-08
    Jean-Daniel Boissonnat, Olivier Devillers, Kunal Dutta, Marc Glisse

    Randomized incremental construction (RIC) is one of the most important paradigms for building geometric data structures. Clarkson and Shor developed a general theory that led to numerous algorithms which are both simple and efficient in theory and in practice. Randomized incremental constructions are usually space-optimal and time-optimal in the worst case, as exemplified by the construction of convex

    更新日期:2020-09-08
  • Tverberg-Type Theorems with Altered Intersection Patterns (Nerves)
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-09-08
    Jesús A. De Loera, Thomas A. Hogan, Deborah Oliveros, Dominic Yang

    Tverberg’s theorem says that a set with sufficiently many points in \({\mathbb {R}}^d\) can always be partitioned into m parts so that the \((m-1)\)-simplex is the (nerve) intersection pattern of the convex hulls of the parts. The main results of our paper demonstrate that Tverberg’s theorem is just a special case of a more general situation, where other simplicial complexes must always arise as nerve

    更新日期:2020-09-08
  • Smallest k -Enclosing Rectangle Revisited
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-09-02
    Timothy M. Chan, Sariel Har-Peled

    Given a set of n points in the plane, and a parameter \(k\), we consider the problem of computing the minimum (perimeter or area) axis-aligned rectangle enclosing \(k\) points. We present the first near quadratic time algorithm for this problem, improving over the previous near-\(O(n^{5/2})\)-time algorithm by Kaplan et al. (25th European Symposium on Algorithms. Leibniz Int Proc Inform, vol. 87, # 52

    更新日期:2020-09-02
  • Random Geometric Complexes and Graphs on Riemannian Manifolds in the Thermodynamic Limit
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-08-31
    Antonio Lerario, Raffaella Mulas

    We investigate some topological properties of random geometric complexes and random geometric graphs on Riemannian manifolds in the thermodynamic limit. In particular, for random geometric complexes we prove that the normalized counting measure of connected components, counted according to isotopy type, converges in probability to a deterministic measure. More generally, we also prove similar convergence

    更新日期:2020-09-01
  • On Mutually Diagonal Nets on (Confocal) Quadrics and 3-Dimensional Webs
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-08-31
    Arseniy V. Akopyan, Alexander I. Bobenko, Wolfgang K. Schief, Jan Techter

    Canonical parametrisations of classical confocal coordinate systems are introduced and exploited to construct non-planar analogues of incircular (IC) nets on individual quadrics and systems of confocal quadrics. Intimate connections with classical deformations of quadrics that are isometric along asymptotic lines and circular cross-sections of quadrics are revealed. The existence of octahedral webs

    更新日期:2020-09-01
  • The Combinatorial Geometry of Stresses in Frameworks
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-08-18
    Oleg Karpenkov

    Consider a realization of a graph in the space with straight segments representing edges. Let us assign a stress for every its edge. In case if at every vertex of the graph the stresses sum up to zero, we say that the realization is a tensegrity. Some realizations possess non-zero tensegrities while the others do not. In this paper we study necessary and sufficient existence conditions for tensegrities

    更新日期:2020-08-18
  • The $$h^*$$ h ∗ -Polynomials of Locally Anti-Blocking Lattice Polytopes and Their $$\gamma $$ γ -Positivity
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-08-12
    Hidefumi Ohsugi, Akiyoshi Tsuchiya

    A lattice polytope \(\mathscr {P} \subset \mathbb {R}^d\) is called a locally anti-blocking polytope if for any closed orthant \({\mathbb R}^d_{\varepsilon }\) in \(\mathbb {R}^d\), \(\mathscr {P} \cap \mathbb {R}^d_{\varepsilon }\) is unimodularly equivalent to an anti-blocking polytope by reflections of coordinate hyperplanes. We give a formula for the \(h^*\)-polynomials of locally anti-blocking

    更新日期:2020-08-12
  • Geometric Multicut: Shortest Fences for Separating Groups of Objects in the Plane
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-08-11
    Mikkel Abrahamsen, Panos Giannopoulos, Maarten Löffler, Günter Rote

    We study the following separation problem: Given a collection of pairwise disjoint coloured objects in the plane with k different colours, compute a shortest “fence” F, i.e., a union of curves of minimum total length, that separates every pair of objects of different colours. Two objects are separated if F contains a simple closed curve that has one object in the interior and the other in the exterior

    更新日期:2020-08-11
  • Local Conditions for Triangulating Submanifolds of Euclidean Space
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-08-10
    Jean-Daniel Boissonnat, Ramsay Dyer, Arijit Ghosh, Andre Lieutier, Mathijs Wintraecken

    We consider the following setting: suppose that we are given a manifold M in \({\mathbb {R}}^d\) with positive reach. Moreover assume that we have an embedded simplical complex \({\mathcal {A}}\) without boundary, whose vertex set lies on the manifold, is sufficiently dense and such that all simplices in \({\mathcal {A}}\) have sufficient quality. We prove that if, locally, interiors of the projection

    更新日期:2020-08-11
  • A Spanner for the Day After
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-08-06
    Kevin Buchin, Sariel Har-Peled, Dániel Oláh

    We show how to construct a \((1+\varepsilon )\)-spanner over a set \({P}\) of n points in \({\mathbb {R}}^d\) that is resilient to a catastrophic failure of nodes. Specifically, for prescribed parameters \({\vartheta },\varepsilon \in (0,1)\), the computed spanner \({G}\) has $$\begin{aligned} {{\mathcal {O}}}\bigl (\varepsilon ^{-O(d)} {\vartheta }^{-6} n(\log \log n)^6 \log n \bigr ) \end{aligned}$$

    更新日期:2020-08-06
  • A Fast Shortest Path Algorithm on Terrain-like Graphs
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-08-04
    Vincent Froese, Malte Renken

    Terrain visibility graphs are a well-known graph class in computational geometry. They are closely related to polygon visibility graphs, but a precise graph-theoretical characterization is still unknown. Over the last decade, terrain visibility graphs attracted considerable attention in the context of time series analysis (there called time series visibility graphs) with various practical applications

    更新日期:2020-08-05
  • Computing the Fréchet Gap Distance
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-08-03
    Chenglin Fan, Benjamin Raichel

    Measuring the similarity of two polygonal curves is a fundamental computational task. Among alternatives, the Fréchet distance is one of the most well-studied similarity measures. Informally, the Fréchet distance is described as the minimum leash length required for a man on one of the curves to walk a dog on the other curve continuously from the starting to the ending points. In this paper we study

    更新日期:2020-08-03
  • On Grids in Point-Line Arrangements in the Plane
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-07-29
    Mozhgan Mirzaei, Andrew Suk

    The famous Szemerédi–Trotter theorem states that any arrangement of n points and n lines in the plane determines \(O(n^{4/3})\) incidences, and this bound is tight. In this paper, we prove the following Turán-type result for point-line incidence. Let \(\mathcal {L}_a\) and \(\mathcal {L}_b\) be two sets of t lines in the plane and let \(P=\{\ell _a \cap \ell _b : \ell _a \in \mathcal {L}_a, \,\ell

    更新日期:2020-07-29
  • Dynamic Geometric Data Structures via Shallow Cuttings
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-07-24
    Timothy M. Chan

    We present new results on a number of fundamental problems about dynamic geometric data structures: (1) We describe the first fully dynamic data structures with sublinear amortized update time for maintaining (i) the number of vertices or the volume of the convex hull of a 3D point set, (ii) the largest empty circle for a 2D point set, (iii) the Hausdorff distance between two 2D point sets, (iv) the

    更新日期:2020-07-24
  • Reconstruction of Convex Bodies from Moments
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-07-22
    Astrid Kousholt, Julia Schulte

    We investigate how much information about a convex body can be retrieved from a finite number of its geometric moments. We give a sufficient condition for a convex body to be uniquely determined by a finite number of its geometric moments, and we show that among all convex bodies, those which are uniquely determined by a finite number of moments form a dense set. Further, we derive a stability result

    更新日期:2020-07-23
  • On the Number of Perfect Triangles with a Fixed Angle
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-07-20
    Mehdi Makhul

    Richard Guy asked the following question: can we find a triangle with rational sides, medians and area? Such a triangle is called a perfect triangle and no example has been found to date. It is widely believed that such a triangle does not exist. Here we use the setup of Solymosi and de Zeeuw about rational distance sets contained in an algebraic curve, to show that for any angle \(0<\theta < \pi \)

    更新日期:2020-07-20
  • Eliminating Depth Cycles Among Triangles in Three Dimensions
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-07-14
    Boris Aronov, Edward Y. Miller, Micha Sharir

    The vertical depth relation among n pairwise openly disjoint triangles in 3-space may contain cycles. We show that, for any \(\varepsilon >0\), the triangles can be cut into \(O(n^{3/2+\varepsilon })\) connected semialgebraic pieces, whose description complexity depends only on the choice of \(\varepsilon \), such that the depth relation among these pieces is now a proper partial order. This bound

    更新日期:2020-07-14
  • On Weak $$\epsilon $$ ϵ -Nets and the Radon Number
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-07-13
    Shay Moran, Amir Yehudayoff

    We show that the Radon number characterizes the existence of weak nets in separable convexity spaces (an abstraction of the Euclidean notion of convexity). The construction of weak nets when the Radon number is finite is based on Helly’s property and on metric properties of VC classes. The lower bound on the size of weak nets when the Radon number is large relies on the chromatic number of the Kneser

    更新日期:2020-07-13
  • On the Number of Weakly Connected Subdigraphs in Random k NN Digraphs
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-07-06
    Selim Bahadır, Elvan Ceyhan

    We study the number of copies of a weakly connected subdigraph of the k nearest neighbor (kNN) digraph based on data from certain random point processes in \(\mathbb {R}^d\). In particular, based on the asymptotic theory for functionals of point sets from homogeneous Poisson process (HPP) and uniform binomial process (UBP), we provide a general result for the asymptotic behavior of the number of weakly

    更新日期:2020-07-06
  • Computing Min-Convex Hulls in the Affine Building of $$\hbox {SL}_d$$ SL d
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-07-06
    Leon Zhang

    We describe an algorithm for computing the min-convex hull of a finite collection of points in the affine building of \(\hbox {SL}_d(K)\), for K a field with discrete valuation. These min-convex hulls describe the relations among a finite collection of invertible matrices over K. As a consequence, we bound the dimension of the tropical projective space needed to realize the min-convex hull as a tropical

    更新日期:2020-07-06
  • Finding Needles in a Haystack
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-07-06
    Árpád Kurusa

    Convex polygons are distinguishable among the piecewise \(C^\infty \) convex domains by comparing their visual angle functions on any surrounding circle. This is a consequence of our main result, that every segment in a \(C^\infty \) multicurve can be reconstructed from the masking function of the multicurve given on any surrounding circle.

    更新日期:2020-07-06
  • The Graphs Behind Reuleaux Polyhedra
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-07-06
    Luis Montejano, Eric Pauli, Miguel Raggi, Edgardo Roldán-Pensado

    This work is about graphs arising from Reuleaux polyhedra. Such graphs must necessarily be planar, 3-connected and strongly self-dual. We study the question of when these conditions are sufficient. If G is any such graph, each vertex has an opposite face with isomorphism \(\tau :G \rightarrow G^*\) (where \(G^*\) is the unique dual graph), a metric mapping is a map \(\eta :V(G) \rightarrow \mathbb

    更新日期:2020-07-06
  • Near-Optimal Algorithms for Shortest Paths in Weighted Unit-Disk Graphs
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-06-24
    Haitao Wang, Jie Xue

    We revisit a classical graph-theoretic problem, the single-source shortest-path (SSSP) problem, in weighted unit-disk graphs. We first propose an exact (and deterministic) algorithm which solves the problem in \(O(n\log ^2\!n)\) time using linear space, where n is the number of the vertices of the graph. This significantly improves the previous deterministic algorithm by Cabello and Jejčič [CGTA’15]

    更新日期:2020-06-25
  • Constructing Planar Support for Non-Piercing Regions
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-06-22
    Rajiv Raman, Saurabh Ray

    Given a hypergraph \(\mathcal {H}=(X,{\mathcal {S}})\), a planar support for \(\mathcal {H}\) is a planar graph G with vertex set X, such that for each hyperedge \(S\in \mathcal {S}\), the subgraph of G induced by the vertices in S is connected. Planar supports for hypergraphs have found several algorithmic applications, including several packing and covering problems, hypergraph coloring, and in hypergraph

    更新日期:2020-06-23
  • The Schläfli Fan
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-06-22
    Michael Joswig, Marta Panizzut, Bernd Sturmfels

    Smooth tropical cubic surfaces are parametrized by maximal cones in the unimodular secondary fan of the triple tetrahedron. There are \(344\, 843 \,867\) such cones, organized into a database of \(14\,373\,645\) symmetry classes. The Schläfli fan gives a further refinement of these cones. It reveals all possible patterns of lines on tropical cubic surfaces, thus serving as a combinatorial base space

    更新日期:2020-06-23
  • Theorems of Carathéodory, Helly, and Tverberg Without Dimension
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-06-19
    Karim Adiprasito, Imre Bárány, Nabil H. Mustafa, Tamás Terpai

    We initiate the study of no-dimensional versions of classical theorems in convexity. One example is Carathéodory’s theorem without dimension: given an n-element set P in a Euclidean space, a point \(a \in {{\,\mathrm{{\texttt {conv}}}\,}}P\), and an integer \(r \le n\), there is a subset \(Q\subset P\) of r elements such that the distance between a and \({{\,\mathrm{{\texttt {conv}}}\,}}Q\) is less

    更新日期:2020-06-19
  • Discrete Equidecomposability and Ehrhart Theory of Polygons
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-06-10
    Paxton Turner, Yuhuai Wu

    Motivated by questions from Ehrhart theory, we present new results on discrete equidecomposability. Two rational polygons P and Q are said to be discretely equidecomposable if there exists a piecewise affine-unimodular bijection (equivalently, a piecewise affine-linear bijection that preserves the integer lattice \({\mathbb {Z}}^2\)) from P to Q. We develop an invariant for a particular version of

    更新日期:2020-06-10
  • Almost All String Graphs are Intersection Graphs of Plane Convex Sets
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-06-05
    János Pach, Bruce Reed, Yelena Yuditsky

    A string graph is the intersection graph of a family of continuous arcs in the plane. The intersection graph of a family of plane convex sets is a string graph, but not all string graphs can be obtained in this way. We prove the following structure theorem conjectured by Janson and Uzzell: The vertex set of almost all string graphs on n vertices can be partitioned into five cliques such that some pair

    更新日期:2020-06-05
  • An Exploration of Locally Spherical Regular Hypertopes
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-06-03
    Maria Elisa Fernandes, Dimitri Leemans, Asia Ivić Weiss

    Hypertope is a generalization of the concept of polytope, which in turn generalizes the concept of a map and hypermap, to higher rank objects. Regular hypertopes with spherical residues, which we call regular locally spherical hypertopes, must be either of spherical, euclidean, or hyperbolic type. That is, the type-preserving automorphism group of a locally spherical regular hypertope is a quotient

    更新日期:2020-06-03
  • Intersection Patterns of Planar Sets
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-06-02
    Gil Kalai, Zuzana Patáková

    Let \({\mathcal {A}}=\{A_1,\ldots ,A_n\}\) be a family of sets in the plane. For \(0 \le i < n\), denote by \(f_i\) the number of subsets \(\sigma \) of \(\{1,\ldots ,n\}\) of cardinality \(i+1\) that satisfy \(\bigcap _{i \in \sigma } A_i \ne \emptyset \). Let \(k \ge 2\) be an integer. We prove that if each k-wise and \((k{+}1)\)-wise intersection of sets from \({\mathcal {A}}\) is empty, or a single

    更新日期:2020-06-02
  • Sheaf-Theoretic Stratification Learning from Geometric and Topological Perspectives
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-05-29
    Adam Brown, Bei Wang

    We investigate a sheaf-theoretic interpretation of stratification learning from geometric and topological perspectives. Our main result is the construction of stratification learning algorithms framed in terms of a sheaf on a partially ordered set with the Alexandroff topology. We prove that the resulting decomposition is the unique minimal stratification for which the strata are homogeneous and the

    更新日期:2020-05-29
  • On the Number of Monochromatic Lines in $$\pmb {\mathbb {R}}^d$$Rd
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-05-27
    Mario Huicochea

    Let X be a nonempty finite subset of \({\mathbb {R}}^d\) and \(X=\bigcup _{i=1}^m X_i\) a coloring with \(m

    更新日期:2020-05-27
  • Symmetric Non-Negative Forms and Sums of Squares
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-05-21
    Grigoriy Blekherman, Cordian Riener

    We study symmetric non-negative forms and their relationship with symmetric sums of squares. For a fixed number of variables n and degree 2d, symmetric non-negative forms and symmetric sums of squares form closed, convex cones in the vector space of n-variate symmetric forms of degree 2d. Using representation theory of the symmetric group we characterize both cones in a uniform way. Further, we investigate

    更新日期:2020-05-21
  • Admissible Complexes for the Projective X-ray Transform over a Finite Field
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-05-09
    David V. Feldman, Eric L. Grinberg

    We consider the X-ray transform in a projective space over a finite field. It is well known (after Bolker) that this transform is injective. We formulate an analog of Gelfand’s admissibility problem for the Radon transform, which asks for a classification of all minimal sets of lines for which the restricted Radon transform is injective. The solution involves doubly ruled quadric surfaces.

    更新日期:2020-05-09
  • Simple Realizability of Complete Abstract Topological Graphs Simplified
    Discret. Comput. Geom. (IF 0.621) Pub Date : 2020-05-04
    Jan Kynčl

    An abstract topological graph (briefly an AT-graph) is a pair \(A=(G,{\mathcal {X}})\) where \(G=(V,E)\) is a graph and \({\mathcal {X}}\subseteq {E \atopwithdelims ()2}\) is a set of pairs of its edges. The AT-graph A is simply realizable if G can be drawn in the plane so that each pair of edges from \({\mathcal {X}}\) crosses exactly once and no other pair crosses. We show that simply realizable

    更新日期:2020-05-04
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