The convex dimension of a graph G = (V, E) is the smallest dimension d for which G admits an injective map f : V −→ R of its vertices into d-space, such that the barycenters of the images of the edges of G are in convex position. The strong convex dimension of G is the smallest d for which G admits a map as above such that the images of the vertices of G are also in convex position. In this pap...

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 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 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.

Given a positive and an increasing nonlinearity f that satisfies an appropriate growth condition at infinity, we provide a condition on g ∈ C∞(Ω) for which the Monge-Ampère equation detD2u = gf(u) admits a solution with infinite boundary value on a strictly convex domain Ω. Sufficient conditions for the nonexistence of such solutions will also be given.

We prove a local existence and uniqueness result of crystalline mean curvature flow starting from a compact convex admissible set in R . This theorem can handle the facet breaking/bending phenomena, and can be generalized to any anisotropic mean curvature flow. The method provides also a generalized geometric evolution starting from any compact convex set, existing up to the extinction time, sa...

Small sets. We are going to present some results of the following type: under a suitable assumption on a normed space X, each continuous convex function, defined on an open convex set A ⊂ X, is (Gâteaux or Fréchet) differentiable outside a set which is in some sense small. It is natural to ask that a nonempty family S ⊂ 2X whose elements are considered “small sets” satisfy the following conditi...