Radon Numbers Grow Linearly

Define 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 .

On (L,M)-fuzzy convex structures

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

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).

Decomposable Convexities in Graphs and Hypergraphs

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).

Generalised Helly and Radon numbers

At the Second Oklahoma Conference on Convexity and Related Combinatorial Geometry (1980) G.Sierksma posed several problems dealing with the generalised Helly and Radon numbers of a convexity space. The aim of this note is to give answers to and comment on some of Sierksma's questions.