Properties of the metric projection on weakly vial-convex sets and parametrization of set-valued mappings with weakly convex images

We continue studying the class of weakly convex sets (in the sense of Vial). For points in a sufficiently small neighborhood of a closed weakly convex subset in Hubert space, we prove that the metric projection on this set exists and is unique. In other words, we show that the closed weakly convex sets have a Chebyshev layer. We prove that the metric projection of a point on a weakly convex set satisfies the Lipschitz condition with respect to a point and the Hölder condition with exponent 1/2 with respect to a set. We develop a method for constructing a continuous parametrization of a set-valued mapping with weakly convex images. We obtain an explicit estimate for the modulus of continuity of the parametrizing function.     

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.     

Convergence of decreasing sequences of convex sets in nonreflexive Banach spaces

We consider nested sequences of linear or convex closed sets of the form arising in estimation and other inverse problems. We show that such sequences may fail to converge in any of the recently studied set convergences other than Mosco convergence. We also provide a positive result concerning the epislice convergence of related sequences of functions.Research partially supported by NSERC operating grants.     

Directional End of a Convex Set: Theory and Applications

A point of a convex set belongs to its end in a given direction when this direction is not feasible at that point. This paper analyzes the properties of the directional end of general convex sets and closed convex sets (for which the directional ends are connected by arcs) as well as the relationship between the directional end and certain concepts on the illumination of convex bodies. The paper includes applications of the directional end to the theory of linear systems.     

A note on non-support points, negligible sets, Gâteaux differentiability and Lipschitz embeddings

This paper first presents a characterization of three classes of negligible closed convex sets (i.e., Gauss null sets, Aronszajn null sets and cube null sets) in terms of non-support points; then gives a generalization of Gâteaux differentiability theorems of Lipschitz mapping from open sets to those closed convex sets admitting non-support points; and as their application, finally shows that a closed convex set in a separable Banach space X can be Lipschitz embedded into a Banach space Y with the Radon–Nikodym property if and only if the closure of its linear span is linearly isomorphic to a closed subspace of Y.     

闭凸集上的一类非线性半变分不等式解的存在性

考虑一类定义在闭凸集上的非线性半变分不等式问题,通过运用闭凸集上的临界点理论、Clarke次微分性质以及非光滑紧性条件等,得到了这类半变分不等式解的存在性．     

An Unsolved Problem of Fenchel

The Fenchel problem of level sets is solved under the conditions that theboundaries of the nested family of convex sets in R^n>+1 aregiven by C³ n-dimensional differentiable manifolds and theconvex sets determine an open or closed convex set inRⁿ⁺¹.     

Strong Unicity of Restricted p-Centers

Given a continuous seminorm p on a separated locally convex linear topological space X, we study in this paper strong uniqueness of the so-called restricted p-centers of sets. First we explore finite extremal characterizations of strongly unique restricted p-centers of subsets of X from its finite dimensional subspaces. Our main goal here is to investigate strong uniqueness of restricted p-centers of certain sets from the so-called p-RS sets, which are defined as closed convex sets that are obtained by imposing convex side constraints arising from boundedness of coefficients on translates of certain subspaces of X.     

Hilbertian convex feasibility problem: Convergence of projection methods

The classical problem of finding a point in the intersection of countably many closed and convex sets in a Hilbert space is considered. Extrapolated iterations of convex combinations of approximate projections onto subfamilies of sets are investigated to solve this problem. General hypotheses are made on the regularity of the sets and various strategies are considered to control the order in which the sets are selected. Weak and strong convergence results are established within thisbroad framework, which provides a unified view of projection methods for solving hilbertian convex feasibility problems. This work was supported by the National Science Foundation under Grant MIP-9308609.     

Approximation of the limit distance function in Banach spaces

In this paper we study the behavior of the limit distance function d(x)=limdist(x,Cn) defined by a nested sequence (Cn) of subsets of a real Banach space X. We first present some new criteria for the non-emptiness of the intersection of a nested sequence of sets and of their ε-neighborhoods from which we derive, among other results, Dilworth's characterization [S.J. Dilworth, Intersections of centred sets in normed spaces, Far East J. Math. Sci. (Part II) (1988) 129-136 (special volume)] of Banach spaces not containing c₀ and Marino's result [G. Marino, A remark on intersection of convex sets, J. Math. Anal. Appl. 284 (2003) 775-778]. Passing then to the approximation of the limit distance function, we show three types of results: (i) that the limit distance function defined by a nested sequence of non-empty bounded closed convex sets coincides with the distance function to the intersection of the weak^∗-closures in the bidual; this extends and improves the results in [J.M.F. Castillo, P.L. Papini, Distance types in Banach spaces, Set-Valued Anal. 7 (1999) 101-115]; (ii) that the convexity condition is necessary; and (iii) that in spaces with separable dual, the distance function to a weak^∗-compact convex set is approximable by a (non-necessarily nested) sequence of bounded closed convex sets of the space.     

On convex bodies of constant width

We present an alternative proof of the following fact: the hyperspace of compact closed subsets of constant width in Rn is a contractible Hilbert cube manifold. The proof also works for certain subspaces of compact convex sets of constant width as well as for the pairs of compact convex sets of constant relative width. Besides, it is proved that the projection map of compact closed subsets of constant width is not 0-soft in the sense of Shchepin, in particular, is not open.     

Convex functions with continuous epigraph or continuous level sets

Convex functions with continuous epigraph in the sense of Gale and Klée have been studied recently by Auslender and Coutat in a finite-dimensional setting. Here, we provide characterizations of such functionals in terms of the Legendre-Fenchel transformation in general locally convex spaces. Also, we show that the concept of continuous convex sets is of interest in these spaces. We end with a characterization of convex functions on Euclidean spaces with continuous level sets.     

A sequential iteration algorithm with non-monotoneous behaviour in the method of projections onto convex sets

The method of projections onto convex sets to find a point in the intersection of a finite number of closed convex sets in a Euclidean space, may lead to slow convergence of the constructed sequence when that sequence enters some narrow “corridor” between two or more convex sets. A way to leave such corridor consists in taking a big step at different moments during the iteration, because in that way the monotoneous behaviour that is responsible for the slow convergence may be interrupted. In this paper we present a technique that may introduce interruption of the monotony for a sequential algorithm, but that at the same time guarantees convergence of the constructed sequence to a point in the intersection of the sets. We compare experimentally the behaviour concerning the speed of convergence of the new algorithm with that of an existing monotoneous algorithm.     

On Finite Linear Systems Containing Strict Inequalities

This paper deals with linear systems containing finitely many weak and/or strict inequalities, whose solution sets are referred to as evenly convex polyhedral sets. The classical Motzkin theorem states that every (closed and convex) polyhedron is the Minkowski sum of a convex hull of finitely many points and a finitely generated cone. In this sense, similar representations for evenly convex polyhedra have been recently given by using the standard version for classical polyhedra. In this work, we provide a new dual tool that completely characterizes finite linear systems containing strict inequalities and it constitutes the key for obtaining a generalization of Motzkin theorem for evenly convex polyhedra.     

Distance characterizations of finite lattices

Semimodular, modular and distributive finite lattices are characterized by means of convex sublattices and distance closed sets of the Hasse diagram graphs.     

On the range sets of variational inequalities

This paper investigates the closedness and convexity of the range sets of the variational inequality (VI) problem defined by an affine mappingM and a nonempty closed convex setK. It is proved that the range set is closed ifK is the union of a polyhedron and a compact convex set. Counterexamples are given such that the range set is not closed even ifK is a simple geometrical figure such as a circular cone or a circular cylinder in a three-dimensional space. Several sufficient conditions for closedness and convexity of the range set are presented. Characterization for the convex hull of the range set is established in the case whereK is a cone, while characterization for the closure of the convex hull of the range set is established in general. Finally, some applications to stability of VI problems are derived.This work was supported by the Australian Research Council.We are grateful to Professors M. Seetharama Gowda, Olvi Mangasarian, Jong-Shi Pang, and Steve Robinson for references. We are thankful to Professor Jim Burke for discussions on Theorem 2.1 and Counterexample 3.5.     

Analytical Linear Inequality Systems and Optimization

In many interesting semi-infinite programming problems, all the constraints are linear inequalities whose coefficients are analytical functions of a one-dimensional parameter. This paper shows that significant geometrical information on the feasible set of these problems can be obtained directly from the given coefficient functions. One of these geometrical properties gives rise to a general purification scheme for linear semi-infinite programs equipped with so-called analytical constraint systems. It is also shown that the solution sets of such kind of consistent systems form a transition class between polyhedral convex sets and closed convex sets in the Euclidean space of the unknowns.     

Extrapolation algorithm for affine-convex feasibility problems

The convex feasibility problem under consideration is to find a common point of a countable family of closed affine subspaces and convex sets in a Hilbert space. To solve such problems, we propose a general parallel block-iterative algorithmic framework in which the affine subspaces are exploited to introduce extrapolated over-relaxations. This framework encompasses a wide range of projection, subgradient projection, proximal, and fixed point methods encountered in various branches of applied mathematics. The asymptotic behavior of the method is investigated and numerical experiments are provided to illustrate the benefits of the extrapolations.     

Some properties of convex hulls of integer points contained in general convex sets

In this paper, we study properties of general closed convex sets that determine the closedness 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 classes of convex sets such as pointed cones, strictly convex sets, and sets containing integer points in their interior. We then present a sufficient condition for the convex hull of integer points in general convex sets to be a polyhedron. This result generalizes the well-known result due to Meyer (Math Program 7:223–225, 1974). Under a simple technical assumption, we show that these sufficient conditions are also necessary for the convex hull of integer points contained in general convex sets to be a polyhedron.     

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.