Article history: Received 30 October 2015 Accepted 25 July 2016 Available online 28 July 2016 Submitted by B. Lemmens MSC: 90C53 15A09

This paper discusses under what conditions two disjoint convex subsets of a linear topological space can be separated by a continuous linear functional. The equivalence of several forms of the Hahn-Banach theorem is proven. The separation problem is considered in linear topological spaces, locally convex linear topological spaces, Banach spaces, and finally finite dimensional Banach spaces. A n...

We study the problem of finding small weak ε-nets in three dimensions and provide new upper and lower bounds on the value of ε for which a weak ε-net of a given small constant size exists. The range spaces under consideration are the set of all convex sets and the set of all halfspaces in R3.

In this paper, we establish some new results related to distributive properties of AND ( )  and OR ( )  -operations with respect to operations of union, restricted union restricted intersection, extended intersection and restricted difference on soft sets, and provide some illustrative examples.

A finite collection C of closed convex sets in R is said to have a (p, q)-property if among any p members of C some q have a non-empty intersection, and |C| ≥ p. A piercing number of C is defined as the minimal number k such that there exists a k-element set which intersects every member of C. We focus on the simplest non-trivial case in R, i.e., p = 4 and q = 3. It is known that the maximum po...

The Definition of a Convex Set In Rd, a set S of points is convex if the line segment joining any two points of S lies completely within S (Figure 1). The purpose of this article is to describe a recent extension of this concept of convexity to the Grassmannian and to discuss its connection with some other ideas in geometry. More specifically, the extension is to the so-called “affine Grassmann...

θ 1 + · · · + θ k = 1. Show that θ 1 x 1 + · · · + θ k x k ∈ C. (The definition of convexity is that this holds for k = 2; you must show it for arbitrary k.) Hint. Use induction on k. Solution. This is readily shown by induction from the definition of convex set. We illustrate the idea for k = 3, leaving the general case to the reader. Suppose that x 1 , x 2 , x 3 ∈ C, and θ 1 + θ 2 + θ 3 = 1 w...

In this paper, we consider a very useful and significant class of convex sets and convex functions that is relative convex sets and relative convex functions which was introduced and studied by Noor [20]. Several new inequalities of Hermite-Hadamard type for relative convex functions are established using different approaches. We also introduce relative h-convex functions and is shown that rela...

Let P be a set of n points in general position in the plane. A subset I of P is called an island if there exists a convex set C such that I = P ∩C. In this paper we define the generalized island Johnson graph of P as the graph whose vertex consists of all islands of P of cardinality k, two of which are adjacent if their intersection consists of exactly l elements. We show that for large enough ...

We consider three operators which appear naturally in convexity theory and determine completely the structure of the semigroup generated by them.