Convexity Space with Respect to a Given Set

1998 ◽  
pp. 199-208 ◽  
Author(s):  
Lucia Blaga ◽  
Liana Lupşa
Keyword(s):  
Convexity Space

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).

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.

scholarly journals Extending a convexity space to an aligned space

1984 ◽  
Vol 87 (4) ◽  
pp. 429-435 ◽  
Author(s):  
Gerard Sierksma
Keyword(s):  
Convexity Space

scholarly journals Radon Numbers Grow Linearly

2021 ◽  
Author(s):  
Dömötör Pálvölgyi
Keyword(s):  
Convex Hulls ◽  
Convexity Space ◽  
Helly Theorem ◽  
Radon Number

AbstractDefine the k-th Radon number $$r_k$$ r k of a convexity space as the smallest number (if it exists) for which any set of $$r_k$$ r k points can be partitioned into k parts whose convex hulls intersect. Combining the recent abstract fractional Helly theorem of Holmsen and Lee with earlier methods of Bukh, we prove that $$r_k$$ r k grows linearly, i.e., $$r_k\le c(r_2)\cdot k$$ r k ≤ c ( r 2 ) · k .

An Internal Solution to the Problem of Linearization of a Convexity Space

1976 ◽  
Vol 19 (4) ◽  
pp. 487-494 ◽  
Author(s):  
D. A. Szafron ◽  
J. H. Weston
Keyword(s):  
Topological Space ◽  
Convex Sets ◽  
Sufficient Conditions ◽  
Linear Structure ◽  
Linear Topological Space ◽  
Necessary And Sufficient Conditions ◽  
Convexity Space ◽  
Internal Solution ◽  
Convexity Structure ◽  
Necessary And Sufficient

Following Kay and Womble [2] an abstract convexity structure on a set X is a collection ξ of subsets of X which includes the empty set, X and is closed under arbitrary intersections. One of the natural problems that arises in convexity structures is to give necessary and sufficient conditions for the existance of a linear structure on X such that the collection of all convex sets in the resulting linear space is precisely ξ. An associated problem is to consider a set with a convexity structure and a topology and find necessary and sufficient conditions for the existance of a linear structure on X such that X becomes a linear topological space with again ξ the collection of convex sets.

The starshapedness number and a Krasnosel'skii-type theorem in a convexity space

1987 ◽  
Vol 49 (6) ◽  
pp. 535-544 ◽  
Author(s):  
Krzysztof Kołodziejczyk
Keyword(s):  
Type Theorem ◽  
Convexity Space

scholarly journals On (L,M)-fuzzy convex structures

Filomat ◽  
2019 ◽  
Vol 33 (13) ◽  
pp. 4151-4163 ◽  
Author(s):  
Osama Sayed ◽  
El-Sayed El-Sanousy ◽  
Yaser Sayed
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Topology ◽  
Convexity Space ◽  
New Class

This paper defines a new class of L-fuzzy sets called r-L-fuzzy biconvex sets in (L,M)-fuzzy convex structures (X,C), where C is an (L,M)-fuzzy convexity on X, and some of their properties were studied. In addition, weintroduce (L,M)-fuzzy topological convexity space and study some of its properties. Finally, we introduce locally (L,M)-fuzzy topology (L,M)-fuzzy convexity space and study some of its properties.

A new notion of convexity in digraphs with an application to Bayesian networks

2017 ◽  
Vol 09 (02) ◽  
pp. 1750016
Author(s):  
Francesco M. Malvestuto
Keyword(s):  
Bayesian Networks ◽  
Bayesian Network ◽  
Convex Geometry ◽  
Convexity Space ◽  
Linear Algorithms ◽  
Markov Properties

We introduce a new notion of convexity in digraphs, which we call incoming-path convexity, and prove that the incoming-path convexity space of a digraph is a convex geometry (that is, it satisfies the Minkowski–Krein–Milman property) if and only if the digraph is acyclic. Moreover, we prove that incoming-path convexity is adequate to characterize collapsibility of models generated by Bayesian networks. Based on these results, we also provide simple linear algorithms to solve two topical problems on Markov properties of a Bayesian network (that is, on conditional independences valid in a Bayesian network).

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