If F is a vector space over an ordered division ring, C a convex subset of V and L the lattice of convex subsets of C, then we call L a convexity lattice. We give necessary and sufficient conditions for an abstract lattice to be a convexity lattice in the finite dimensional case.

It is proven that the intersection of convex transversals of finitely many sets in R is a convex polytope, possibly empty. Related results on support hyperplanes and complements of finite unions of convex sets are given. MSC 2000: 52A20

We show that every in nite dimensional Banach space has a closed and bounded convex set that is not remotal. This settles a problem raised by Sababheh and Khalil in [8].

When a single group uses majority rule to select a set of policies from an n-dimensional compact and convex set, a core generally exists if and only if n = 1. Finding analogous conditions for core existence when an n-dimensional action requires agreement from m groups has been an open problem. This paper provides a solution to this problem by establishing sufficient conditions for core existenc...

We give three proofs of the fact that a smoothly bounded, convex domain in R n has defining functions whose Hessians are non-negative definite in a neighborhood of the boundary of the domain.

We describe the shape of the Binomial distribution, especially its convexity.

We show that the number of unit-area triangles determined by a set S of n points in the plane is O(n), improving the earlier bound O(n) of Apfelbaum and Sharir [2]. We also consider two special cases of this problem: (i) We show, using a somewhat subtle construction, that if S consists of points on three lines, the number of unit-area triangles that S spans, with one vertex on each of the lines...

The function hK is called the support function of K because it provides information on support hyperplanes and halfspaces of K . Indeed, every support halfspace of K is of the form {x :x · u≤ hK(u)} for some u ∈ Sd−1 and since K equals the intersection of all its support halfspaces, the function hK uniquely determines K . For a proof of this and other elementary properties of the support functi...

We show that, given a closed convex set K with the origin in its interior, the support function of the set {y ∈ K∗ | ∃x ∈ K such that xy = 1} is the pointwise smallest sublinear function σ such that K = {x |σ(x) ≤ 1}.