A Tverberg-type generalization of the Helly number of a convexity space

1981 ◽  
Vol 16 (1) ◽  
pp. 117-125 ◽  
Author(s):  
Jean-Paul Doignon ◽  
John R. Reay ◽  
Gerard Sierksma
Keyword(s):  
Convexity Space ◽  
Helly Number

On the dual Helly number and irreducible decompositions of the unit in lattices

1993 ◽  
Vol 54 (3) ◽  
pp. 899-902
Author(s):  
A. P. Zolotarev
Keyword(s):  
Helly Number

Remark on the Helly number for strongly convex sets on Riemannian manifolds

1981 ◽  
Vol 34 (1) ◽  
pp. 27-29 ◽  
Author(s):  
Norbert Kleinjohann
Keyword(s):  
Riemannian Manifolds ◽  
Convex Sets ◽  
Helly Number ◽  
Strongly Convex

scholarly journals Decomposable Convexities in Graphs and Hypergraphs

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Francesco M. Malvestuto
Keyword(s):  
Convex Hull ◽  
Convex Sets ◽  
Nonempty Subset ◽  
Convex Set ◽  
Convex Geometry ◽  
Convexity Space ◽  
Vertex Set ◽  
Convex Geometries ◽  
Acyclic Hypergraph ◽  
Maximal Clusters

Given a connected hypergraph with vertex set V, a convexity space on is a subset of the powerset of V that contains ∅, V, and the singletons; furthermore, is closed under intersection and every set in is connected in . The members of are called convex sets. The convex hull of a subset X of V is the smallest convex set containing X. By a cluster of we mean any nonempty subset of V in which every two vertices are separated by no convex set. We say that a convexity space on is decomposable if it satisfies the following three axioms: (i) the maximal clusters of form an acyclic hypergraph, (ii) every maximal cluster of is a convex set, and (iii) for every nonempty vertex set X, a vertex does not belong to the convex hull of X if and only if it is separated from X by a convex cluster. We prove that a decomposable convexity space on is fully specified by the maximal clusters of in that (1) there is a closed formula which expresses the convex hull of a set in terms of certain convex clusters of and (2) is a convex geometry if and only if the subspaces of induced by maximal clusters of are all convex geometries. Finally, we prove the decomposability of some known convexities in graphs and hypergraphs taken from the literature (such as “monophonic” and “canonical” convexities in hypergraphs and “all-paths” convexity in graphs).

A helly-number fork-almost-neighborly sets

1972 ◽  
Vol 11 (4) ◽  
pp. 347-348 ◽  
Author(s):  
Marilyn Breen
Keyword(s):  
Helly Number

scholarly journals A helly number for unions of two boxes inR2

1985 ◽  
Vol 8 (2) ◽  
pp. 267-273 ◽  
Author(s):  
Marilyn Breen
Keyword(s):  
Convex Sets ◽  
Helly Number ◽  
Boundary Points ◽  
Polygonal Region

LetSbe a polygonal region in the plane with edges parallel to the coordinate axes. If every5or fewer boundary points ofScan be partitioned into setsAandBso thatconv A⋃ conv B⫅S, thenSis a union of two convex sets, each a rectangle. The number5is best possible.Without suitable hypothesis on edges ofS, the theorem fails. Moreover, an example reveals that there is no finite Helly number which characterizes arbitrary unions of two convex sets, even for polygonal regions in the plane.

scholarly journals On the Computational Complexity of the Helly Number in the P3 and Related Convexities

2019 ◽  
Vol 346 ◽  
pp. 285-297
Author(s):  
Moisés T. Carvalho ◽  
Simone Dantas ◽  
Mitre C. Dourado ◽  
Daniel F.D. Posner ◽  
Jayme L. Szwarcfiter
Keyword(s):  
Computational Complexity ◽  
Helly Number

Linearization of a Convexity Space

1976 ◽  
Vol s2-13 (2) ◽  
pp. 209-214 ◽  
Author(s):  
P. Mah ◽  
S. A. Naimpally ◽  
J. H. M. Whitfield
Keyword(s):  
Convexity Space

scholarly journals Topological convexity spaces

1974 ◽  
Vol 19 (2) ◽  
pp. 125-132 ◽  
Author(s):  
Victor Bryant
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Topological Properties ◽  
Convexity Space ◽  
Basic Properties

We shall start by recalling the definition and some basic properties of a convexity space; a topological convexity space (tcs) will then be a convexity space together with an admissible topology, and will be a generalisation of a topological vector space (tvs). After showing that the usual tvs results connecting the linear and topological properties extend to this new setting we then prove a form of the Krein-Milman theorem in a tcs.

Maximal $S$-Free Convex Sets and the Helly Number

2016 ◽  
Vol 30 (4) ◽  
pp. 2206-2216 ◽  
Author(s):  
Michele Conforti ◽  
Marco Di Summa
Keyword(s):  
Convex Sets ◽  
Helly Number
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