A condition for a compact plane set to be a union of finitely many convex sets

1974 ◽  
Vol 76 (1) ◽  
pp. 61-66 ◽  
Author(s):  
H. G. Eggleston
Keyword(s):  
Euclidean Plane ◽  
Convex Sets ◽  
The Real ◽  
Compact Plane ◽  
Convex Subsets

All the sets with which we are concerned are subsets of the real Euclidean plane E2. By Lm we denote those subsets X of E2 for which, if pl, p2, …, Pm are any m points of X, then at least one segment pipj, i ≠ j consists entirely of points of X. L2 is the class of convex subsets of E2. We shall show that if X is closed and X ∈ Lm. then X is the union of finitely many convex sets. This extends a result of Valentine (4). See also (1),(2),(3).

scholarly journals Hyperspaces of max-plus convex subsets of powers of the real line

2012 ◽  
Vol 394 (2) ◽  
pp. 481-487 ◽  
Author(s):  
Lidia Bazylevych ◽  
Dušan Repovš ◽  
Mykhailo Zarichnyi
Keyword(s):  
Real Line ◽  
The Real ◽  
Convex Subsets

Porosity and the bounded linear regularity property

2014 ◽  
Vol 20 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Simeon Reich ◽  
Alexander J. Zaslavski
Keyword(s):  
Hilbert Space ◽  
Convex Sets ◽  
Regularity Property ◽  
Nonempty Intersection ◽  
Normed Spaces ◽  
Infinite Products ◽  
Bounded Linear ◽  
Linear Regularity ◽  
Nearest Point ◽  
Convex Subsets

Abstract.H. H. Bauschke and J. M. Borwein showed that in the space of all tuples of bounded, closed, and convex subsets of a Hilbert space with a nonempty intersection, a typical tuple has the bounded linear regularity property. This property is important because it leads to the convergence of infinite products of the corresponding nearest point projections to a point in the intersection. In the present paper we show that the subset of all tuples possessing the bounded linear regularity property has a porous complement. Moreover, our result is established in all normed spaces and for tuples of closed and convex sets, which are not necessarily bounded.

scholarly journals Algebraic models of the Euclidean plane

2018 ◽  
Vol Volume 2 ◽  
Author(s):  
Jérémy Blanc ◽  
Adrien Dubouloz
Keyword(s):  
Euclidean Plane ◽  
Kodaira Dimension ◽  
Algebraic Surfaces ◽  
Algebraic Models ◽  
The Real ◽  
Homology Groups ◽  
Real Algebraic Surfaces ◽  
Compact Case ◽  
Birational Diffeomorphism

We introduce a new invariant, the real (logarithmic)-Kodaira dimension, that allows to distinguish smooth real algebraic surfaces up to birational diffeomorphism. As an application, we construct infinite families of smooth rational real algebraic surfaces with trivial homology groups, whose real loci are diffeomorphic to $\mathbb{R}^2$, but which are pairwise not birationally diffeomorphic. There are thus infinitely many non-trivial models of the euclidean plane, contrary to the compact case. Comment: 16 pages

The Structure of C*-Convex Sets

1994 ◽  
Vol 46 (5) ◽  
pp. 1007-1026 ◽  
Author(s):  
Phillip B. Morenz
Keyword(s):  
Convex Sets ◽  
Extreme Points ◽  
Structural Elements ◽  
The Right ◽  
The Relationship ◽  
Carathéodory Theorem ◽  
Convex Subsets

AbstractCompact C*-convex subsets of Mn correspond exactly to n-th matrix ranges of operators. The main result of this paper is to discover the “right” analog of linear extreme points, called structural elements, and then to prove a generalised Krein-Milman theorem for C*-convex subsets of Mn. The relationship between structural elements and an earlier attempted generalisation, called C*-extreme points, is examined, solving affirmatively a conjecture of Loebl and Paulsen [8]. An improved bound for a C* -convex version of the Caratheodory theorem for convex sets is also given.

Touching Convex Sets in the Plane

1994 ◽  
Vol 37 (4) ◽  
pp. 495-504 ◽  
Author(s):  
Meir Katchalski ◽  
János Pach
Keyword(s):  
Positive Constant ◽  
Euclidean Plane ◽  
Convex Sets ◽  
Nonempty Intersection ◽  
The Other ◽  
Plane Convex ◽  
Straight Line

AbstractTwo subsets of the Euclidean plane touch each other if they have a point in common and there is a straight line separating one from the other.It is shown that there exists a positive constant c such that if are families of plane convex sets with for some k ≥ 1 and if every touches every then either contains k members having nonempty intersection.

scholarly journals SUBLATTICES OF LATTICES OF ORDER-CONVEX SETS, II.

2003 ◽  
Vol 13 (05) ◽  
pp. 543-564 ◽  
Author(s):  
MARINA SEMENOVA ◽  
FRIEDRICH WEHRUNG
Keyword(s):  
Positive Integer ◽  
Partially Ordered Set ◽  
Convex Sets ◽  
Ordered Set ◽  
Finitely Based ◽  
Locally Finite ◽  
Partially Ordered ◽  
Finitely Based Variety ◽  
Atomistic Lattice ◽  
Convex Subsets

For a positive integer n, we denote by SUB (respectively, SUBn) the class of all lattices that can be embedded into the lattice Co(P) of all order-convex subsets of a partially ordered set P (respectively, P of length at most n). We prove the following results: (1) SUBn is a finitely based variety, for any n≥1. (2) SUB2 is locally finite. (3) A finite atomistic lattice L without D-cycles belongs to SUB if and only if it belongs to SUB2; this result does not extend to the nonatomistic case. (4) SUBn is not locally finite for n≥3.

On Certain Intersection Properties of Convex Sets

1951 ◽  
Vol 3 ◽  
pp. 272-275 ◽  
Author(s):  
V. L. Klee
Keyword(s):  
Euclidean Space ◽  
Interior Point ◽  
Convex Sets ◽  
Common Interior Point ◽  
Convex Subsets

A collection of n + 1 convex subsets of a Euclidean space E will be called an n-set in E provided each n of the sets have a common interior point although the intersection of all n + 1 interiors is empty. It is well-known that if {C0,C1} is a 1-set, then C0 and C1 can be separated by a hyperplane.

Extreme points of unbounded, closed and convex sets in Banach spaces

1984 ◽  
Vol 95 (2) ◽  
pp. 319-323 ◽  
Author(s):  
Ioannis A. Polyrakis
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Sets ◽  
Extreme Points ◽  
Convex Subsets

In this paper we examine the existence of extreme points in unbounded, closed and convex subsets K of a Banach space X.

A Non-Euclidean Distance

1971 ◽  
Vol 64 (7) ◽  
pp. 595-600
Author(s):  
Stanley R. Clemens
Keyword(s):  
Distance Function ◽  
Euclidean Distance ◽  
Euclidean Plane ◽  
Plane Geometry ◽  
Synthetic Approach ◽  
Real Numbers ◽  
Angle Measure ◽  
The Real ◽  
Measure Function

There are basically two approaches to classical Euclidean plane geometry—the synthetic approach and the metric approach. The older of the two is the synthetic approach followed by Eucliding later by Hilbert. In the Eucliding treatment, one begin by assuming as undefined the relations of betweenness, congruence of segments, and congruence of angles. The metric treatments, initiated by G. D. Birkhoff in the 1930s, assumes the existence of the real numbers (or a set of postulates that guarantees the existence of the real numbers) and the existence of a distance function d and an angle-measure function m.

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