This paper looks at axioms for convexity, and shows how they can be applied to discrete spaces. Two structures for a discrete geometry are considered: oriented matroids, and cell complexes. Oriented matroids are shown to have a structure which naturally satisfies the axioms for being a convex geometry. Cell complexes are shown to give rise to various different notions of convexity, one of which...

Consider a measurable space with a finite vector measure. This measure defines a mapping of the σ-field into a Euclidean space. According to Lyapunov’s convexity theorem, the range of this mapping is compact and, if the measure is atomless, this range is convex. Similar ranges are also defined for measurable subsets of the space. We show that the union of the ranges of all subsets having the sa...

A restricted-orientation convex set is a set of points whose intersection with lines from some fixed set is empty or connected. This notion generalizes both standard convexity and orthogonal convexity. We explore basic properties of restricted-orientation convex sets in three dimensions. In particular, we establish analogs of the following properties of standard convex sets: The intersection of...

The well-known Carath&odory’s theorem says that, given a set VG$” and a point a E conv V (the convex hull of V), there exists a subset A c V such that IAl s n + 1 and a E conv A. This simple theorem has many applications and generalizations (see, for instance, [4, 9, 1OJ). The aim of this paper is to give a new generalization of Carathiodory’s theorem and to present some consequences of thig ge...

The aim of this paper is to establish some new Hermite–Hadamard type inequalities for harmonic h-convex functions involving hypergeometric functions. We also discuss some new and known special cases, which can be deduced from our results. The ideas and techniques of this paper may inspire further research in this field. In recent years, much attention have been given to theory of convexity beca...

A convex body R in a normed d-dimensional space M is called reduced if the M-thickness ∆(K) of each convex body K ⊂ R different from R is smaller than ∆(R). We present two characterizations of reduced polytopes in M. One of them is that a convex polytope P ⊂ M is reduced if and only if through every vertex v of P a hyperplane strictly supporting P passes such that the M-width of P in the perpen...

A restricted-orientation convex set is a set of points whose intersection with lines from some fixed set is empty or connected. This notion generalizes both standard convexity and orthogonal convexity. We explore basic properties of restricted-orientation convex sets in three dimensions. In particular, we establish analogs of the following properties of standard convex sets: The intersection of...

A known result in combinatorial geometry states that any collection Pn of points on the plane contains two such that any circle containing them contains n/c elements of Pn , c a constant. We prove: Let 8 be a family of n noncrossing compact convex sets on the plane, and let S be a strictly convex compact set. Then there are two elements Si , Sj of 8 such that any set S′ homothetic to S that con...

The inaccessibility of a point p in a bounded domain D ⊂ R is the minimum of the lengths of segments through p with boundary at ∂D. The points of maximum inaccessibility ID are those where the inaccessibility achieves its maximum. We prove that for strictly convex domains, ID is either a point or a segment, and that for a planar polygon ID is in general a point. We study the case of a triangle,...

By using parameterized representation of fuzzy numbers, criteria for a lower semicontinuous fuzzy mapping defined on a non-empty convex subset of Rn to be a convex fuzzy mapping are obtained. c © 2007 Elsevier Ltd. All rights reserved.