On closedness of convex sets in Banach lattices

Positivity - Let X be a Banach lattice. A well-known problem arising from the theory of risk measures asks when order closedness of a convex set in X implies closedness with respect to the topology...     

On closedness of law-invariant convex sets in rearrangement invariant spaces

Archiv der Mathematik - This paper presents relations between several types of closedness of a law-invariant convex set in a rearrangement invariant space $${mathcal {X}}$$. In particular, we show...     

On convergence of closed convex sets

In this paper we introduce a convergence concept for closed convex subsets of a finite-dimensional normed vector space. This convergence is called C-convergence. It is defined by appropriate notions of upper and lower limits. We compare this convergence with the well-known Painlevé-Kuratowski convergence and with scalar convergence. In fact, we show that a sequence (An)_n∈NC-converges to A if and only if the corresponding support functions converge pointwise, except at relative boundary points of the domain of the support function of A, to the support function of A.     

On the illumination of unbounded closed convex sets

In this note we prove that the illumination of an almost bounded closed convex set by minimum number of affine subspaces of given dimension can be reduced to the illumination of a bounded closed convex set of lower dimension. The work was supported by Hung. Nat. Found. for Sci. Research No. 326-0213     

On inclusion and summands of bounded closed convex sets

Summary This article introduces coset extensions and group coextensions of S-sets.     

On closed convex sets without boundary rays and asymptotes

We study in finite-dimensional spaces the class of closed convex sets without boundary rays and asymptotes, denoted by and introduced by D. Gale and V. Klee. These sets, not necessarily bounded, enjoy many properties satisfied by compacts sets. New properties of this class are given and convergence analysis of this class is investigated. We also introduce the class of closed convex proper functions which have an epigraph in and we give some properties of these functions.     

Motzkin decomposition of closed convex sets

Theodore Motzkin proved, in 1936, that any polyhedral convex set can be expressed as the (Minkowski) sum of a polytope and a polyhedral convex cone. This paper provides five characterizations of the larger class of closed convex sets in finite dimensional Euclidean spaces which are the sum of a compact convex set with a closed convex cone. These characterizations involve different types of representations of closed convex sets as the support functions, dual cones and linear systems whose relationships are also analyzed in the paper. The obtaining of information about a given closed convex set F and the parametric linear optimization problem with feasible set F from each of its different representations, including the Motzkin decomposition, is also discussed.     

Strict approximation on closed convex sets

The concept of strict approximation over subspaces of an euclidean space, introduced by John R. Rice, is extended to closed convex sets. It is proved that the best p-approximants converge as p→∞ to the strict approximant not generally but when the closed convex set satisfies certain approximative property. Finally, a similar problem is considered in the space c₀ of real sequences tending to 0. This paper was partially supported by the Consojo de Investigacions Cient/ficas y Tecnológicas de la Provincia de Córdoba.     

On the size of the algebraic difference of two random Cantor sets

In this paper we consider some families of random Cantor sets on the line and investigate the question whether the condition that the sum of Hausdorff dimension is larger than one implies the existence of interior points in the difference set of two independent copies. We prove that this is the case for the so called Mandelbrot percolation. On the other hand the same is not always true if we apply a slightly more general construction of random Cantor sets. We also present a complete solution for the deterministic case. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Minimum recession-compatible subsets of closed convex sets

A subset B of a closed convex set A is recession-compatible with respect to A if A can be expressed as the Minkowski sum of B and the recession cone of A. We show that if A contains no line, then there exists a recession-compatible subset of A that is minimal with respect to set inclusion. The proof only uses basic facts of convex analysis and does not depend on Zorn’s Lemma. An application of this result to the error bound theory in optimization is presented.     

Convergence on closed convex sets in a locally convex space

The purpose of this paper is to compare several kinds of convergences on the space C(X) of nonempty closed convex subsets of a locally convex space X. First we verify that the AW-convergence on C(X) is weaker than the metric Attouch-Wets convergence on C(X) of a metrizable locally convex space X. Moreover, we show that X is normable if and only if the two convergences on C(X × R) are equivalent. Secondly we define two convergences on C(X) analogous to the corresponding ones in a normed linear space, and investigate some basic properties of these convergences and compare them.     

Proper holomorphic discs avoiding closed convex sets

Denote by the open unit disc in . Let C be a closed convex subset of . We prove that for each there is a proper holomorphic map such that and if and only if either C is a complex line or C does not contain any complex line. Received: 17 July 2001; in final form: 22 November 2001 / Published online: 5 September 2002