Some properties of convex hulls of integer points contained in general convex sets

Convexity and Geodesic Metric Spaces

In this paper, we first present a preliminary study on metric segments and geodesics in metric spaces. Then we recall the concept of d-convexity of sets and functions in the sense of Menger and study some properties of d-convex sets and d-convex functions as well as extreme points and faces of d-convex sets in normed spaces. Finally we study the continuity of d-convex functions in geodesic metr...

Transversals to the convex hulls of all k-sets of discrete subsets of Rn

Let k, d, λ ≥ 1 be integers with d ≥ λ. What is the maximum positive integer n such that every set of n points in R has the property that the convex hulls of all k-sets have a transversal (d−λ)-plane? What is the minimum positive integer n such that every set of n points in general position in R has the property that the convex hulls of all k-sets do not have a transversal (d− λ)-plane? In this...

Some Properties of Certain Subclasses of Close-to-Convex and Quasi-convex Functions with Respect to 2k-Symmetric Conjugate Points

Sweep Line Algorithm for Convex Hull Revisited

Convex hull of some given points is the intersection of all convex sets containing them. It is used as primary structure in many other problems in computational geometry and other areas like image processing, model identification, geographical data systems, and triangular computation of a set of points and so on. Computing the convex hull of a set of point is one of the most fundamental and imp...

Intersection cuts for nonlinear integer programming: convexification techniques for structured sets

We study the generalization of split and intersection cuts from Mixed Integer Linear Programming to the realm of Mixed Integer Nonlinear Programming. Constructing such cuts requires calculating the convex hull of the difference of two convex sets with specific geometric structures. We introduce two techniques to give precise characterizations of such convex hulls and use them to construct split...