We consider a variant of the standard Arrow{Debreu model which contains a number of indivisible goods and one perfectly divisible good (numeraire or money). The objective of this paper is to clarify a general mathematical mechanism which guarantees the existence of an equilibrium in such a model. It is shown that the crucial condition for the existence of an equilibrium is that the sets of supp...

Restricted-orientation convexity is the study of geometric objects whose intersection with lines from some fixed set is empty or connected. We have studied the properties of restricted-orientation convex sets and demonstrated that this notion is a generalization of standard convexity. We now describe a restricted-orientation generalization of halfspaces and explore properties of these generaliz...

In this paper we present several results on the expected complexity of a convex hull of n points chosen uniformly and independently from a convex shape. (i) We show that the expected number of vertices of the convex hull of n points, chosen uniformly and independently from a disk is O(n1/3), and O(k log n) for the case a convex polygon with k sides. Those results are well known (see [RS63, Ray7...

Strong visibility is a generalization of standard visibility, defined with respect to a fixed set of line orientations. We investigate computational properties of this generalized visibility, as well as the related notion of strong convexity. In particular, we describe algorithms for the following tasks: 1. Testing the strong visibility of two points in a polygon and the strong convexity of a p...

We study the existence and non-existence of fundamental solutions for the scalar conservation laws ut + f(u)x = 0, related to convexity assumptions on f . We also study the limits of those solutions as the initial mass goes to infinity. We especially prove the existence of so-called Friendly Giants and Infinite Shock Solutions according to the convexity of f , which generalize the explicit powe...

Caffarelli-Friedman [7] proved a constant rank theorem for convex solutions of semilinear elliptic equations in R2, a similar result was also discovered by Yau [28] at the same time. The result in [7] was generalized to R by Korevaar-Lewis [27] shortly after. This type of constant rank theorem is called microscopic convexity principle. It is a powerful tool in the study of geometric properties ...

Convexity correction arises when one computes the expected value of an interest rate index under a probability measure other than its own natural martingale measure. As a typical example, the natural martingale measure of the swap rate is the swap measure with annuity as the numeraire. However, the evaluation of the discounted expectation of the payoff in a constant maturity swap (CMS) derivati...

Let R+ = (0,∞) and let M be the family of all mean values of two numbers in R+ (some examples are the arithmetic, geometric, and harmonic means). Given m1, m2 ∈ M, we say that a function f : R+ → R+ is (m1, m2)-convex if f(m1(x, y)) ≤ m2(f(x), f(y)) for all x, y ∈ R+. The usual convexity is the special case when both mean values are arithmetic means. We study the dependence of (m1, m2)-convexit...