A Note on Convexly Independent Subsets of an Infinite Set of Points

2002 ◽  
Vol 9 (2) ◽  
pp. 303-307
Author(s):  
A. Kharazishvili
Keyword(s):  
Euclidean Space ◽  
Point Sets ◽  
Set Of Points ◽  
Infinite Set

Abstract We consider convexly independent subsets of a given infinite set of points in the plane (Euclidean space) and evaluate the cardinality of such subsets. It is demonstrated, in particular, that situations are essentially different for countable and uncountable point sets.

The number of non-homogeneous lattice points in n-dimensional point sets

1953 ◽  
Vol 49 (1) ◽  
pp. 156-157 ◽  
Author(s):  
D. B. Sawyer
Keyword(s):  
Lower Bound ◽  
Euclidean Space ◽  
Lattice Points ◽  
Dimensional Euclidean Space ◽  
Infinite Volume ◽  
Point Sets ◽  
Homogeneous Lattice ◽  
Set Of Points

Let R be a set of points in n-dimensional Euclidean space, and let Δ′(R) denote the lower bound of the determinants of non-homogeneous lattices which have no point in R. For Δ′(R) to be non-zero it is necessary, as Macbeath has shown (2), that R should have infinite volume.

scholarly journals MAINTAINING VISIBILITY INFORMATION OF PLANAR POINT SETS WITH A MOVING VIEWPOINT

2007 ◽  
Vol 17 (04) ◽  
pp. 297-304 ◽  
Author(s):  
OLIVIER DEVILLERS ◽  
VIDA DUJMOVIĆ ◽  
HAZEL EVERETT ◽  
SAMUEL HORNUS ◽  
SUE WHITESIDES ◽  
...  
Keyword(s):  
Linear Space ◽  
Point Sets ◽  
Worst Case ◽  
Planar Point ◽  
Set Of Points

Given a set of n points in the plane, we consider the problem of computing the circular ordering of the points about a viewpoint q and efficiently maintaining this ordering information as q moves. In linear space, and after O(n log n) preprocessing time, our solution maintains the view at a cost of O( log n) amortized time (resp.O( log 2 n) worst case time) for each change. Our algorithm can also be used to maintain the set of points sorted according to their distance to q .

scholarly journals A Helly-type theorem on a sphere

1967 ◽  
Vol 7 (3) ◽  
pp. 323-326 ◽  
Author(s):  
M. J. C. Baker
Keyword(s):  
Euclidean Space ◽  
Convex Sets ◽  
Type Theorem ◽  
Dimensional Euclidean Space ◽  
Dimensional Sphere ◽  
Strongly Convex ◽  
Set Of Points

The purpose of this paper is to prove that if n+3, or more, strongly convex sets on an n dimensional sphere are such that each intersection of n+2 of them is empty, then the intersection of some n+1 of them is empty. (The n dimensional sphere is understood to be the set of points in n+1 dimensional Euclidean space satisfying x21+x22+ …+x2n+1 = 1.)

ON THE STEINER RATIO IN $\mathcal{R}_{n}$

2011 ◽  
Vol 03 (04) ◽  
pp. 473-489
Author(s):  
HAI DU ◽  
WEILI WU ◽  
ZAIXIN LU ◽  
YINFENG XU
Keyword(s):  
Euclidean Space ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Dimensional Euclidean Space ◽  
Minimum Length ◽  
Steiner Ratio ◽  
Line Segments ◽  
Finite Set ◽  
Steiner Minimum Tree ◽  
Set Of Points

The Steiner minimum tree and the minimum spanning tree are two important problems in combinatorial optimization. Let P denote a finite set of points, called terminals, in the Euclidean space. A Steiner minimum tree of P, denoted by SMT(P), is a network with minimum length to interconnect all terminals, and a minimum spanning tree of P, denoted by MST(P), is also a minimum network interconnecting all the points in P, however, subject to the constraint that all the line segments in it have to terminate at terminals. Therefore, SMT(P) may contain points not in P, but MST(P) cannot contain such kind of points. Let [Formula: see text] denote the n-dimensional Euclidean space. The Steiner ratio in [Formula: see text] is defined to be [Formula: see text], where Ls(P) and Lm(P), respectively, denote lengths of a Steiner minimum tree and a minimum spanning tree of P. The best previously known lower bound for [Formula: see text] in the literature is 0.615. In this paper, we show that [Formula: see text] for any n ≥ 2.

scholarly journals On the complexity of some partition problems of a finite set of points in Euclidean space into balanced clusters

2019 ◽  
Vol 488 (1) ◽  
pp. 16-20
Author(s):  
A. V. Kel’manov ◽  
A. V. Pyatkin ◽  
V. I. Khandeev
Keyword(s):  
Euclidean Space ◽  
Cluster Size ◽  
Np Completeness ◽  
Finite Set ◽  
Set Of Points

We consider some problems of partitioning a finite set of N points in d-dimension Euclidean space into two clusters balancing the value of (1) the quadratic variance normalized by a cluster size, (2) the quadratic variance, and (3) the size-weighted quadratic variance. We have proved the NP-completeness of all these problems.

Minkowski spacetime

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Euclidean Space ◽  
Special Relativity ◽  
Three Dimensional ◽  
Minkowski Spacetime ◽  
Dimensional Euclidean Space ◽  
Fourth Dimension ◽  
Relativistic Dynamics ◽  
Newtonian Theory ◽  
Space And Time ◽  
Set Of Points

This chapter presents the main features of the Minkowski spacetime, which is the geometrical framework in which the laws of relativistic dynamics are formulated. It is a very simple mathematical extension of three-dimensional Euclidean space. In special relativity, ‘relative, apparent, and common’ (in the words of Newton) space and time are represented by a mathematical set of points called events, which constitute the Minkowski spacetime. This chapter also stresses the interpretation of the fourth dimension, which in special relativity is time. Here, time now loses the ‘universal’ and ‘absolute’ nature that it had in the Newtonian theory.

Almost Continuity of Mappings

1968 ◽  
Vol 11 (3) ◽  
pp. 453-455 ◽  
Author(s):  
Shwu-Yeng T. Lin
Keyword(s):  
Euclidean Space ◽  
Topological Space ◽  
Direct Proof ◽  
Baire Space ◽  
Real Numbers ◽  
Range Space ◽  
Natural Context ◽  
Set Of Points

Let E be a metric Baire space and f a real valued function on E. Then the set of points of almost continuity in E is dense (everywhere) in E.Our purpose is to set this result in its most natural context, relax some very restricted hypotheses, and to supply a direct proof. More precisely, we shall prove that the metrizability of E in Theorem H may be removed, and that the range space may be generalized from the (Euclidean) space of real numbers to any topological space satisfying the second axiom of countability [2].

scholarly journals Bounding the Size of an Almost-Equidistant Set in Euclidean Space

2018 ◽  
Vol 28 (2) ◽  
pp. 280-286 ◽  
Author(s):  
ANDREY KUPAVSKII ◽  
NABIL H. MUSTAFA ◽  
KONRAD J. SWANEPOEL
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Set Of Points

A set of points in d-dimensional Euclidean space is almost equidistant if, among any three points of the set, some two are at distance 1. We show that an almost-equidistant set in ℝd has cardinality O(d4/3).

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