Let X be a Banach space whose characteristic of noncompact convexity is less than 1 and satisfies the nonstrict Opial condition. Let C be a bounded closed convex subset of X , KC(C) the family of all compact convex subsets of C, and T a nonexpansive mapping from C into KC(C). We prove that T has a fixed point. The nonstrict Opial condition can be removed if, in addition, T is a 1-χcontractive m...

(1.1) det(D2u) = 1 in Rn must be a quadratic polynomial. For n = 2, a classical solution is either convex or concave; the result holds without the convexity hypothesis. A simpler and more analytical proof, along the lines of affine geometry, of the theorem was later given by Cheng and Yau [9]. The first author extended the result for classical solutions to viscosity solutions [4]. It was proven...

Let K be a nonempty compact convex subset of a uniformly convex Banach space, and   K K T   : a multivalued nonexpansive mapping. We prove that the sequences of Noor iterate converge to a fixed point of T. This generalizes former results proved by Banach convergence of Noor iterates for a multi-valued mapping with a fixed point. We also introduce both of the iterative processes in a new sen...

This theorem is similar to the well known result that any point in the convex hull of a set E in an w-dimensional euclidean space lies in the convex hull of some subset of E containing at most n-\-l points [l, 2 ] . 1 In these theorems the set E is an arbitrary set in the space. The convex hull of E, denoted by H(E), is the set product of all convex sets in the space which contain E. A euclidea...

Let M be a hyperbolic 3–manifold, namely a complete 3–dimensional Riemannian manifold of constant curvature −1, such that the fundamental group π1 (M) is finitely generated. A fundamental subset of M is its convex core CM , defined as the smallest non-empty closed convex subset ofM . The boundary ∂CM of this convex core is a surface of finite topological type, and its geometry was described by ...

To a convex set in a Banach space we associate a convex function (the separating function), whose subdifferential provides useful information on the nature of the supporting and exposed points of the convex set. These points are shown to be also connected to the solutions of a minimization problem involving the separating function. We investigate some relevant properties of this function and of...

In a recent preprint by Amaral & Letchford (2006) convex hulls of sets of matrices corresponding to permutations and path-metrics are studied. A symmetric n × n-matrix is a path metric, if there exist points x1, . . . , xn ∈ R such that the matrix entries are just the pairwise distances |xk − xl| between the points and if these distances are at least one whenever k 6= l. The convex hull of the ...

Motivated by typical questions from computational geometry (visibility and art gallery problems) and combinatorial geometry (illumination problems) we present an analogue of the Krein-Milman theorem for the class of star-shaped sets. If S ⊆ R is compact and star-shaped, we consider a fixed, nonempty, compact, and convex subset K of the convex kernel K0 = ck(S) of S, for instance K = K0 itself. ...

Kuhn-Tucker condition (0, 0) e dK(x, y) into a more explicit and familiar form. Writing where -k(y) = /0(x) + ylfl(x] + • • • + ymfm(x) and P is the nonnegative orthant in R, we see from the example following Theorem 20 that the condition 0 e dyK(x, y) holds if and only if the vector On the other hand, for y e P we have by Theorem 20, assuming for example that f{ is a continuous function for i ...

where g:(0, 1)-»X is an essentially bounded strongly measurable function. In this paper we examine analogues of the Radon-Nikodym Property for quasiBanach spaces. If 0 < p < 1, there are several possible ways of defining "differentiable" operators on Lp, but they inevitably lead to the conclusion that the only differentiable operator is zero. For example, a differentiable operator on L\ has the...