The infimal convolution of M-convex functions is M-convex. This is a fundamental fact in discrete convex analysis that is often useful in its application to mathematical economics and game theory. M-convexity and its variant called M-convexity are closely related to gross substitutability, and the infimal convolution operation corresponds to an aggregation. This note provides a succinct descrip...

Let S be a finite set with n elements in a real linear space. Let JS be a set of n intervals in R. We introduce a convex operator co(S,JS) which generalizes the familiar concepts of the convex hull conv S and the affine hull aff S of S. We establish basic properties of this operator. It is proved that each homothet of conv S that is contained in aff S can be obtained using this operator. A vari...

In this paper, we study properties of general closed convex sets that determine the closed-ness and polyhedrality of the convex hull of integer points contained in it. We first present necessary and sufficient conditions for the convex hull of integer points contained in a general convex set to be closed. This leads to useful results for special class of convex sets such as pointed cones, stric...

We investigate the extremal points of a functional ∫ f(∇u), for a convex or concave function f . The admissible functions u : Ω ⊂ RN → R are convex themselves and satisfy a condition u2 ≤ u ≤ u1. We show that the extremal points are exactly u1 and u2 if these functions are convex and coincide on the boundary ∂Ω. No explicit regularity condition is imposed on f , u1, or u2. Subsequently we discu...

In this note we prove that, if the cost function satisfies some necessary structural conditions and the densities are bounded away from zero and infinity, then strictly c-convex potentials arising in optimal transportation belong to W 2,1+κ loc for some κ > 0. This generalizes some recents results [10, 11, 24] concerning the regularity of strictly convex Alexandrov solutions of the Monge-Ampère...

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...

In this paper, new Hermite-Hadamard type inequalities for coordinated convex and co-ordinated quasi convex functions are proved in a unique way. These results generalize many results proved in earlier works for these classes of functions. Finally, applications of our results are given to estimate the product of moments of two independent continuous random variables.

In this paper we establish some new inequalities for differentiable functions based on concavity and s-convexity. We also prove several Hadamard-type inequalities for products of two convex and s-convex functions. 2007 Elsevier Inc. All rights reserved.