scholarly journals On convex holes in d-dimensional point sets

2021 ◽  
pp. 1-8
Author(s):  
Boris Bukh ◽  
Ting-Wei Chao ◽  
Ron Holzman
Keyword(s):  
Numerical Analysis ◽  
Upper Bound ◽  
General Position ◽  
Convex Polytope ◽  
Point Sets ◽  
Finite Set ◽  
Basic Version

Abstract Given a finite set $A \subseteq \mathbb{R}^d$ , points $a_1,a_2,\dotsc,a_{\ell} \in A$ form an $\ell$ -hole in A if they are the vertices of a convex polytope, which contains no points of A in its interior. We construct arbitrarily large point sets in general position in $\mathbb{R}^d$ having no holes of size $O(4^dd\log d)$ or more. This improves the previously known upper bound of order $d^{d+o(d)}$ due to Valtr. The basic version of our construction uses a certain type of equidistributed point sets, originating from numerical analysis, known as (t,m,s)-nets or (t,s)-sequences, yielding a bound of $2^{7d}$ . The better bound is obtained using a variant of (t,m,s)-nets, obeying a relaxed equidistribution condition.

Planar point sets with a small number of empty convex polygons

2004 ◽  
Vol 41 (2) ◽  
pp. 243-269 ◽  
Author(s):  
Imre Bárány ◽  
Pável Valtr
Keyword(s):  
Convex Hull ◽  
General Position ◽  
Point Sets ◽  
Convex Polygons ◽  
Planar Point ◽  
Empty Convex ◽  
Finite Set ◽  
Empty Triangles

A subset A of a finite set P of points in the plane is called an empty polygon, if each point of A is a vertex of the convex hull of A and the convex hull of A contains no other points of P. We construct a set of n points in general position in the plane with only ˜1.62n2 empty triangles, ˜1.94n2 empty quadrilaterals, ˜1.02n2 empty pentagons, and ˜0.2n2 empty hexagons.

scholarly journals NOTE ON THE NUMBER OF OBTUSE ANGLES IN POINT SETS

2014 ◽  
Vol 24 (03) ◽  
pp. 177-181 ◽  
Author(s):  
RUY FABILA-MONROY ◽  
CLEMENS HUEMER ◽  
EULÀLIA TRAMUNS
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
General Position ◽  
Crossing Number ◽  
The Other ◽  
Point Sets ◽  
Convex Position ◽  
Minimum Number

In 1979 Conway, Croft, Erdős and Guy proved that every set S of n points in general position in the plane determines at least [Formula: see text] obtuse angles and also presented a special set of n points to show the upper bound [Formula: see text] on the minimum number of obtuse angles among all sets S. We prove that every set S of n points in convex position determines at least [Formula: see text] obtuse angles, hence matching the upper bound (up to sub-cubic terms) in this case. Also on the other side, for point sets with low rectilinear crossing number, the lower bound on the minimum number of obtuse angles is improved.

scholarly journals Graphs with many vertex-disjoint cycles

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Dieter Rautenbach ◽  
Friedrich Regen
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Linear Time ◽  
Simple Extension ◽  
Disjoint Cycles ◽  
Cyclomatic Number ◽  
Finite Set ◽  
Extension Rule ◽  
International Audience ◽  
Vertex Disjoint

Graph Theory International audience We study graphs G in which the maximum number of vertex-disjoint cycles nu(G) is close to the cyclomatic number mu(G), which is a natural upper bound for nu(G). Our main result is the existence of a finite set P(k) of graphs for all k is an element of N-0 such that every 2-connected graph G with mu(G)-nu(G) = k arises by applying a simple extension rule to a graph in P(k). As an algorithmic consequence we describe algorithms calculating minmu(G)-nu(G), k + 1 in linear time for fixed k.

scholarly journals Criteria for Embedded Eigenvalues for Discrete Schrödinger Operators

2019 ◽  
Author(s):  
Wencai Liu
Keyword(s):  
Upper Bound ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Embedded Eigenvalues ◽  
Finite Set ◽  
Discrete Schrödinger Operators ◽  
Sign Type ◽  
Free Operator ◽  
Set Of Points ◽  
Rational Type

Abstract In this paper, we consider discrete Schrödinger operators of the form, $$\begin{equation*} (Hu)(n) = u({n+1})+u({n-1})+V(n)u(n). \end{equation*}$$We view $H$ as a perturbation of the free operator $H_0$, where $(H_0u)(n)= u({n+1})+u({n-1})$. For $H_0$ (no perturbation), $\sigma _{\textrm{ess}}(H_0)=\sigma _{\textrm{ac}}(H)=[-2,2]$ and $H_0$ does not have eigenvalues embedded into $(-2,2)$. It is an interesting and important problem to identify the perturbation such that the operator $H_0+V$ has one eigenvalue (finitely many eigenvalues or countable eigenvalues) embedded into $(-2,2)$. We introduce the almost sign type potentials and develop the Prüfer transformation to address this problem, which leads to the following five results. 1: We obtain the sharp spectral transition for the existence of irrational type eigenvalues or rational type eigenvalues with even denominators.2: Suppose $\limsup _{n\to \infty } n|V(n)|=a<\infty .$ We obtain a lower/upper bound of $a$ such that $H_0+V$ has one rational type eigenvalue with odd denominator.3: We obtain the asymptotical behavior of embedded eigenvalues around the boundaries of $(-2,2)$.4: Given any finite set of points $\{ E_j\}_{j=1}^N$ in $(-2,2)$ with $0\notin \{ E_j\}_{j=1}^N+\{ E_j\}_{j=1}^N$, we construct the explicit potential $V(n)=\frac{O(1)}{1+|n|}$ such that $H=H_0+V$ has eigenvalues $\{ E_j\}_{j=1}^N$.5: Given any countable set of points $\{ E_j\}$ in $(-2,2)$ with $0\notin \{ E_j\}+\{ E_j\}$, and any function $h(n)>0$ going to infinity arbitrarily slowly, we construct the explicit potential $|V(n)|\leq \frac{h(n)}{1+|n|}$ such that $H=H_0+V$ has eigenvalues $\{ E_j\}$.

Second Order Dehn Functions for Amalgamated Free Products of Groups

2009 ◽  
Vol 16 (04) ◽  
pp. 699-708
Author(s):  
Xiaofeng Wang ◽  
Xiaomin Bao
Keyword(s):  
Free Product ◽  
Upper Bound ◽  
Second Order ◽  
Free Products ◽  
Dehn Functions ◽  
Amalgamated Free Products ◽  
Products Of Groups ◽  
Amalgamated Free Product ◽  
Finite Set

A finite set of generators for a free product of two groups of type F3with a subgroup amalgamated, and an estimation for the upper bound of the second order Dehn functions of the amalgamated free product are carried out.

SIMULTANEOUS EDGE FLIPPING IN TRIANGULATIONS

2003 ◽  
Vol 13 (02) ◽  
pp. 113-133 ◽  
Author(s):  
JERÔME GALTIER ◽  
FERRAN HURTADO ◽  
MARC NOY ◽  
STÉPHANE PÉRENNES ◽  
JORGE URRUTIA
Keyword(s):  
Upper Bound ◽  
Point Sets

We generalize the operation of flipping an edge in a triangulation to that of flipping several edges simultaneously. Our main result is an optimal upper bound on the number of simultaneous flips that are needed to transform a triangulation into another. Our results hold for triangulations of point sets and for polygons.

scholarly journals Counting Triangulations of Planar Point Sets

10.37236/557 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Micha Sharir ◽  
Adam Sheffer
Keyword(s):  
Upper Bound ◽  
Planar Graphs ◽  
Spanning Trees ◽  
Upper Bounds ◽  
Line Graphs ◽  
Point Sets ◽  
Planar Point ◽  
Straight Line ◽  
Point Set ◽  
Spanning Cycles

We study the maximal number of triangulations that a planar set of $n$ points can have, and show that it is at most $30^n$. This new bound is achieved by a careful optimization of the charging scheme of Sharir and Welzl (2006), which has led to the previous best upper bound of $43^n$ for the problem. Moreover, this new bound is useful for bounding the number of other types of planar (i.e., crossing-free) straight-line graphs on a given point set. Specifically, it can be used to derive new upper bounds for the number of planar graphs ($207.84^n$), spanning cycles ($O(68.67^n)$), spanning trees ($O(146.69^n)$), and cycle-free graphs ($O(164.17^n)$).

scholarly journals Packing Plane Perfect Matchings into a Point Set

2015 ◽  
Vol Vol. 17 no.2 (Graph Theory) ◽  
Author(s):  
Ahmad Biniaz ◽  
Prosenjit Bose ◽  
Anil Maheshwari ◽  
Michiel Smid
Keyword(s):  
Lower Bound ◽  
General Position ◽  
Perfect Matchings ◽  
Point Sets ◽  
Exact Answer ◽  
International Audience ◽  
Point Set

International audience Given a set $P$ of $n$ points in the plane, where $n$ is even, we consider the following question: How many plane perfect matchings can be packed into $P$? For points in general position we prove the lower bound of &#x230A;log<sub>2</sub>$n$&#x230B;$-1$. For some special configurations of point sets, we give the exact answer. We also consider some restricted variants of this problem.

scholarly journals On the Number of Similar Instances of a Pattern in a Finite Set

10.37236/4972 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Bernardo M. Ábrego ◽  
Silvia Fernández-Merchant ◽  
Daniel J. Katz ◽  
Levon Kolesnikov
Keyword(s):  
Arithmetic Progression ◽  
Finite Subset ◽  
Upper Bound ◽  
Point Subset ◽  
Finite Set ◽  
Equilateral Triangles ◽  
Optimal Configurations ◽  
Full Classification ◽  
General Method

New bounds on the number of similar or directly similar copies of a pattern within a finite subset of the line or the plane are proved. The number of equilateral triangles whose vertices all lie within an $n$-point subset of the plane is shown to be no more than $\lfloor{(4 n-1)(n-1)/18}\rfloor$. The number of $k$-term arithmetic progressions that lie within an $n$-point subset of the line is shown to be at most $(n-r)(n+r-k+1)/(2 k-2)$, where $r$ is the remainder when $n$ is divided by $k-1$. This upper bound is achieved when the $n$ points themselves form an arithmetic progression, but for some values of $k$ and $n$, it can also be achieved for other configurations of the $n$ points, and a full classification of such optimal configurations is given. These results are achieved using a new general method based on ordering relations.

scholarly journals A Tight Lower Bound for Convexly Independent Subsets of the Minkowski Sums of Planar Point Sets

10.37236/484 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Ondřej Bílka ◽  
Kevin Buchin ◽  
Radoslav Fulek ◽  
Masashi Kiyomi ◽  
Yoshio Okamoto ◽  
...  
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Minkowski Sum ◽  
Independent Subset ◽  
Point Sets ◽  
Planar Point ◽  
Minkowski Sums

Recently, Eisenbrand, Pach, Rothvoß, and Sopher studied the function $M(m, n)$, which is the largest cardinality of a convexly independent subset of the Minkowski sum of some planar point sets $P$ and $Q$ with $|P| = m$ and $|Q| = n$. They proved that $M(m,n)=O(m^{2/3}n^{2/3}+m+n)$, and asked whether a superlinear lower bound exists for $M(n,n)$. In this note, we show that their upper bound is the best possible apart from constant factors.

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