scholarly journals The brick polytope of a sorting network

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Vincent Pilaud ◽  
Francisco Santos
Keyword(s):  
Convex Hull ◽  
Convex Polygon ◽  
Minkowski Sum ◽  
Sorting Network ◽  
Pseudoline Arrangement ◽  
International Audience ◽  
Pseudoline Arrangements

International audience The associahedron is a polytope whose graph is the graph of flips on triangulations of a convex polygon. Pseudotriangulations and multitriangulations generalize triangulations in two different ways, which have been unified by Pilaud and Pocchiola in their study of pseudoline arrangements with contacts supported by a given network. In this paper, we construct the "brick polytope'' of a network, obtained as the convex hull of the "brick vectors'' associated to each pseudoline arrangement supported by the network. We characterize its vertices, describe its faces, and decompose it as a Minkowski sum of simpler polytopes. Our brick polytopes include Hohlweg and Lange's many realizations of the associahedron, which arise as brick polytopes of certain well-chosen networks. L'associaèdre est un polytope dont le graphe est le graphe des flips sur les triangulations d'un polygone convexe. Les pseudotriangulations et les multitriangulations généralisent les triangulations dans deux directions différentes, qui ont été unifiées par Pilaud et Pocchiola au travers de leur étude des arrangements de pseudodroites avec contacts couvrant un support donné. Nous construisons ici le "polytope de briques'' d'un support, obtenu comme l'enveloppe convexe des "vecteurs de briques'' associés à chaque arrangement de pseudodroites couvrant ce support. Nous caractérisons les sommets de ce polytope, décrivons ses faces et le décomposons en somme de Minkowski de polytopes élémentaires. Notre construction contient toutes les réalisations de l'associaèdre d'Hohlweg et Lange, qui apparaissent comme polytopes de briques de certains supports bien choisis.

scholarly journals Calculus rules for combinations of ellipsoids and applications

1993 ◽  
Vol 47 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Alberto Seeger
Keyword(s):  
Convex Hull ◽  
Convex Polytope ◽  
Finite Family ◽  
Minkowski Sum ◽  
Calculus Rules ◽  
Ellipsoidal Approximations

We derive formulas for the Minkowski sum, the convex hull, the intersection, and the inverse sum of a finite family of ellipsoids. We show how these formulas can be used to obtain inner and outer ellipsoidal approximations of a convex polytope.

scholarly journals Exact Formulae for Variances of Functionals of Convex Hulls

2013 ◽  
Vol 45 (04) ◽  
pp. 917-924
Author(s):  
Christian Buchta
Keyword(s):  
Convex Hull ◽  
Asymptotic Distribution ◽  
Convex Polygon ◽  
Convex Hulls ◽  
Affine Invariance ◽  
Uniform Sample

The vertices of the convex hull of a uniform sample from the interior of a convex polygon are known to be concentrated close to the vertices of the polygon. Furthermore, the remaining area of the polygon outside of the convex hull is concentrated close to the vertices of the polygon. In order to see what happens in a corner of the polygon given by two adjacent edges, we consider—in view of affine invariance—n points P 1,…, P n distributed independently and uniformly in the interior of the triangle with vertices (0, 1), (0, 0), and (1, 0). The number of vertices of the convex hull, which are close to the origin (0, 0), is then given by the number Ñ n of points among P 1,…, P n , which are vertices of the convex hull of (0, 1), P 1,…, P n , and (1, 0). Correspondingly, D̃ n is defined as the remaining area of the triangle outside of this convex hull. We derive exact (nonasymptotic) formulae for var Ñ n and var . These formulae are in line with asymptotic distribution results in Groeneboom (1988), Nagaev and Khamdamov (1991), and Groeneboom (2012), as well as with recent results in Pardon (2011), (2012).

scholarly journals Triangulations of root polytopes and reduced forms (Extended abstract)

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Karola Mészáros
Keyword(s):  
Convex Hull ◽  
Reduced Form ◽  
Vertex Set ◽  
International Audience ◽  
Reduced Forms ◽  
Root Polytope

International audience The type $A_n$ root polytope $\mathcal{P}(A_n^+)$ is the convex hull in $\mathbb{R}^{n+1}$ of the origin and the points $e_i-e_j$ for $1 \leq i < j \leq n+1$. Given a tree $T$ on vertex set $[n+1]$, the associated root polytope $\mathcal{P}(T)$ is the intersection of $\mathcal{P}(A_n^+)$ with the cone generated by the vectors $e_i-e_j$, where $(i, j) \in E(T)$, $i < j$. The reduced forms of a certain monomial $m[T]$ in commuting variables $x_{ij}$ under the reduction $x_{ij} x_{jk} \to x_{ik} x_{ij} + x_{jk} x_{ik} + \beta x_{ik}$, can be interpreted as triangulations of $\mathcal{P}(T)$. If we allow variables $x_{ij}$ and$x_{kl}$ to commute only when $i, j, k, l$ are distinct, then the reduced form of $m[T]$ is unique and yields a canonical triangulation of $\mathcal{P}(T)$ in which each simplex corresponds to a noncrossing alternating forest. Le polytope des racines $\mathcal{P}(A_n^+)$ de type $A_n$ est l'enveloppe convexe dans $\mathbb{R}^{n+1}$ de l'origine et des points $e_i-e_j$ pour $1 \leq i < j \leq n+1$. Étant donné un arbre $T$ sur l'ensemble des sommets $[n+1]$, le polytope des racines associé, $\mathcal{P}(T)$, est l'intersection de $\mathcal{P}(A_n^+)$ avec le cône engendré par les vecteurs $e_i-e_j$, où $(i, j) \in E(T)$, $i < j$. Les formes réduites d'un certain monôme $m[T]$ en les variables commutatives $x_{ij}$ sous la reduction $x_{ij} x_{jk} \to x_{ik} x_{ij} + x_{jk} x_{ik} + \beta x_{ik}$ peuvent être interprétées comme des triangulations de $\mathcal{P}(T)$. Si on impose la restriction que les variables $x_{ij}$ et $x_{kl}$ commutent seulement lorsque les indices $i, j, k, l$ sont distincts, alors la forme réduite de $m[T]$ est unique et produit une triangulation canonique de $\mathcal{P}(T)$ dans laquelle chaque simplexe correspond à une forêt alternée non croisée.

scholarly journals Three notions of tropical rank for symmetric matrices

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Dustin Cartwright ◽  
Melody Chan
Keyword(s):  
Convex Hull ◽  
Symmetric Matrices ◽  
Star Tree ◽  
Close Study ◽  
International Audience ◽  
Donnent Lieu ◽  
Tropical Varieties

International audience We introduce and study three different notions of tropical rank for symmetric matrices and dissimilarity matrices in terms of minimal decompositions into rank 1 symmetric matrices, star tree matrices, and tree matrices. Our results provide a close study of the tropical secant sets of certain nice tropical varieties, including the tropical Grassmannian. In particular, we determine the dimension of each secant set, the convex hull of the variety, and in most cases, the smallest secant set which is equal to the convex hull. Nous introduisons et étudions trois notions différentes de rang tropical pour des matrices symétriques et des matrices de dissimilarité, en utilisant des décompositions minimales en matrices symétriques de rang 1, en matrices d'arbres étoiles, et en matrices d'arbres. Nos résultats donnent lieu à une étude détaillée des ensembles des sécantes tropicales de certaines jolies variétés tropicales, y compris la grassmannienne tropicale. En particulier, nous déterminons la dimension de chaque ensemble des sécantes, l'enveloppe convexe de la variété, ainsi que, dans la plupart des cas, le plus petit ensemble des sécantes qui est égal à l'enveloppe convexe.

scholarly journals Enumerating Triangulations of Convex Polytopes

2001 ◽  
Vol DMTCS Proceedings vol. AA,... (Proceedings) ◽  
Author(s):  
Sergei Bespamyatnikh
Keyword(s):  
Convex Hull ◽  
Simplicial Complex ◽  
Convex Polytopes ◽  
Three Dimensions ◽  
Finite Point ◽  
Convex Position ◽  
International Audience ◽  
Point Set

International audience A triangulation of a finite point set A in $\mathbb{R}^d$ is a geometric simplicial complex which covers the convex hull of $A$ and whose vertices are points of $A$. We study the graph of triangulations whose vertices represent the triangulations and whose edges represent geometric bistellar flips. The main result of this paper is that the graph of triangulations in three dimensions is connected when the points of $A$ are in convex position. We introduce a tree of triangulations and present an algorithm for enumerating triangulations in $O(log log n)$ time per triangulation.

scholarly journals Joint Distribution of the Number of Vertices and the Area of Convex Hulls Generated by a Uniform Distribution in a Convex Polygon

2021 ◽  
pp. 232-243
Author(s):  
Isakjan M. Khamdamov ◽  
Zoya S. Chay
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Convex Hull ◽  
Uniform Distribution ◽  
Joint Distribution ◽  
Poisson Point Process ◽  
Convex Polygon ◽  
Point Processes ◽  
Poisson Approximation ◽  
Convex Hulls

A convex hull generated by a sample uniformly distributed on the plane is considered in the case when the support of a distribution is a convex polygon. A central limit theorem is proved for the joint distribution of the number of vertices and the area of a convex hull using the Poisson approximation of binomial point processes near the boundary of the support of distribution. Here we apply the results on the joint distribution of the number of vertices and the area of convex hulls generated by the Poisson distribution given in [6]. From the result obtained in the present paper, in particular, follow the results given in [3, 7], when the support is a convex polygon and the convex hull is generated by a homogeneous Poisson point process

scholarly journals Exact Formulae for Variances of Functionals of Convex Hulls

2013 ◽  
Vol 45 (4) ◽  
pp. 917-924 ◽  
Author(s):  
Christian Buchta
Keyword(s):  
Convex Hull ◽  
Asymptotic Distribution ◽  
Convex Polygon ◽  
Convex Hulls ◽  
Affine Invariance ◽  
Uniform Sample

The vertices of the convex hull of a uniform sample from the interior of a convex polygon are known to be concentrated close to the vertices of the polygon. Furthermore, the remaining area of the polygon outside of the convex hull is concentrated close to the vertices of the polygon. In order to see what happens in a corner of the polygon given by two adjacent edges, we consider—in view of affine invariance—n points P1,…, Pn distributed independently and uniformly in the interior of the triangle with vertices (0, 1), (0, 0), and (1, 0). The number of vertices of the convex hull, which are close to the origin (0, 0), is then given by the number Ñn of points among P1,…, Pn, which are vertices of the convex hull of (0, 1), P1,…, Pn, and (1, 0). Correspondingly, D̃n is defined as the remaining area of the triangle outside of this convex hull. We derive exact (nonasymptotic) formulae for var Ñn and var . These formulae are in line with asymptotic distribution results in Groeneboom (1988), Nagaev and Khamdamov (1991), and Groeneboom (2012), as well as with recent results in Pardon (2011), (2012).

A New Method for Accurately Evaluating Planar Straightness Error

2010 ◽  
Vol 37-38 ◽  
pp. 18-22
Author(s):  
Jiang Yuan ◽  
Zi Xue Qiu ◽  
Jin Wei Cao
Keyword(s):  
Computational Geometry ◽  
Convex Hull ◽  
Traditional Method ◽  
High Speed ◽  
Convex Polygon ◽  
Accurate Method ◽  
Unique Point ◽  
Automatic Data ◽  
Slope Factor ◽  
Straightness Error

Because of the disadvantages in evaluating straightness error of the traditional method, such as high algorithm complexity, low evaluating precision, it can’t be appropriate for various instruments. A fast and accurate method called intercept of convex polygon algorithm is put forward to implement least envelope zone for evaluating plane straightness errors. The algorithm is based on the convex-hull theory in computational geometry, the measuring data of various instruments are firstly converted into coordinate value, then constructed the convex polygon by slope factor. The unique point meeting the least condition can be found, and the straightness error can be calculated through shear shift conversion. Experimental results show that the method is simple and easy for automatic data processing by computer, and has features of high precision and high speed.

An Efficient Approach of Convex Hull Triangulation Based on Monotonic Chain

2012 ◽  
Vol 263-266 ◽  
pp. 1605-1608
Author(s):  
Yu Ping Zhang ◽  
Zhao Ri Deng ◽  
Rui Qi Zhang
Keyword(s):  
Convex Hull ◽  
Convex Polygon ◽  
Efficient Approach ◽  
Point Set ◽  
Delaunay Algorithm ◽  
Better Than

The triangulation of convex hull has the characteristics of point-set and polygon triangulation. According to some relative definitions, this paper proposed a triangulation of convex hull based on a monotonic chain. This method is better than Delaunay algorithm and is more efficient than other convex polygon algorithms. It is a good algorithm.

scholarly journals A PRIORI FILTRATION OF POINTS FOR FINDING CONVEX HULL

2006 ◽  
Vol 12 (4) ◽  
pp. 341-346 ◽  
Author(s):  
Laura Vyšniauskaitė ◽  
Vydūnas Šaltenis
Keyword(s):  
Convex Hull ◽  
Convex Polygon ◽  
A Priori ◽  
Divide And Conquer ◽  
Brute Force ◽  
Experiment Research ◽  
Main Attention ◽  
Minimum Area ◽  
Good Efficiency ◽  
New Algorithms

Convex hull is the minimum area convex polygon containing the planar set. By now there are quite many convex hull algorithms (Graham Scan, Jarvis March, QuickHull, Incremental, Divide‐and‐Conquer, Marriage‐before‐Conquest, Monotone Chain, Brute Force). The main attention while choosing the algorithm is paid to the running time. In order to raise the efficiency of all the algorithms an idea of a priori filtration of points is given in this article. Besides, two new algorithms have been created and presented. The experiment research has shown a very good efficiency of these algorithms.

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