On (n,m)-Convex Sets

We investigate the class of generalized convex sets on Grassmann manifolds, which includes known generalizations of convex sets for Euclidean spaces. We extend duality theorems (of polarity type) to a broad class of subsets of the Euclidean space. We establish that the invariance of a mapping on generalized convex sets is equivalent to its affinity.     

Polar Transform of Conformally Flat Metrics

In the theory of convex subsets in a Euclidean space, an important role is played by Minkowski duality (the polar transform of a convex set, or the Legendre transform of a convex set). We consider conformally flat Riemannian metrics on the n-dimensional unit sphere and their embeddings into the isotropic cone of the Lorentz space. For a given class of metrics, we define and carry out a detailed study of the Legendre transform.     

Best proximity points,cyclic Kannan maps and geodesic metric spaces

We introduce the concept of cyclic Kannan orbital C-nonexpansive mappings and obtain the existence of a best proximity point on a pair of bounded, closed and convex subsets of a strictly convex metric space by using the geometric notion of seminormal structure. We also study the structure of minimal sets for cyclic Kannan C-nonexpansive mappings and show that results similar to the celebrated Goebel– Karlovitz lemma for nonexpansive self-mappings can be obtained for cyclic Kannan C-nonexpansive mappings.     

On the strong CE-property of convex sets

We consider a class of convex bounded subsets of a separable Banach space. This class includes all convex compact sets as well as some noncompact sets important in applications. For sets in this class, we obtain a simple criterion for the strong CE-property, i.e., the property that the convex closure of any continuous bounded function is a continuous bounded function. Some results are obtained concerning the extension of functions defined at the extreme points of a set in this class to convex or concave functions defined on the entire set with preservation of closedness and continuity. Some applications of the results in quantum information theory are considered.     

Join-semidistributive lattices and convex geometries

We introduce the notion of a convex geometry extending the notion of a finite closure system with the anti-exchange property known in combinatorics. This notion becomes essential for the different embedding results in the class of join-semidistributive lattices. In particular, we prove that every finite join-semidistributive lattice can be embedded into a lattice S_P(A) of algebraic subsets of a suitable algebraic lattice A. This latter construction, S_P(A), is a key example of a convex geometry that plays an analogous role in hierarchy of join-semidistributive lattices as a lattice of equivalence relations does in the class of modular lattices. We give numerous examples of convex geometries that emerge in different branches of mathematics from geometry to graph theory. We also discuss the introduced notion of a strong convex geometry that might promise the development of rich structural theory of convex geometries.     

On the volume of the convex hull of two convex bodies

In this note we examine the volume of the convex hull of two congruent copies of a convex body in Euclidean $n$ -space, under some subsets of the isometry group of the space. We prove inequalities for this volume if the two bodies are translates, or reflected copies of each other about a common point or a hyperplane containing it. In particular, we give a proof of a related conjecture of Rogers and Shephard.     

On a fixed point result of Amini-Harandi in strictly convex Banach spaces

Summary Amini-Harandi proved that alternate convexically nonexpansive mappings on non-empty weakly compact convex subsets of strictly convex Banach spaces have fixed points. We prove that Amini-Harandi's result holds also in Banach spaces with the Kadec--Klee property and the result is true for a larger class of mappings. Moreover, we show that the Alspach mapping in L¹[0,1] is not a 2-alternate convexically nonexpansive mapping.     

The Kakutani fixed point theorem for Roberts spaces

Roberts spaces were the first examples of compact convex subsets of Hausdorff topological vector spaces (HTVS) where the Krein–Milman theorem fails. Because of this exotic quality they were candidates for a counterexample to Schauder's conjecture: any compact convex subset of a HTVS has the fixed point property. However, extending the notion of admissible subsets in HTVS of Klee [Math. Ann. 141 (1960) 286–296], Ngu [Topology Appl. 68 (1996) 1–12] showed the fixed point property for a class of spaces, including the Roberts spaces, he called weakly admissible spaces. We prove the Kakutani fixed point theorem for this class and apply it to show the non-linear alternative for weakly admissible spaces.     

D.C. Representability of closed sets in reflexive Banach spaces and applications to optimization problems

A closed subsetM of a Hausdorff locally convex space is called d.c. representable if there are an extended-real valued lsc convex functionf and a continuous convex functionh such that $$M = \{ x \in X:f(x) - h(x) \leqslant 0\} .$$ Using the existence of a locally uniformly convex norm, we prove that any closed subset in a reflexive Banach space is d.c. representable. For d.c. representable subsets, we define an index of nonconvexity, which can be regarded as an indicator for the degree of nonconvexity. In fact, we show that a convex closed subset is weakly closed when it has a finite index of nonconvexity, and optimization problems on closed subsets with a low index of nonconvexity are less difficult from the viewpoint of computation.     

Sets in a euclidean space which are not a countable union of convex subsets

We prove that if a closed planar setS is not a countable union of convex subsets, then exactly one of the following holds:

Convex compactness and its applications

The concept of convex compactness, weaker than the classical notion of compactness, is introduced and discussed. It is shown that a large class of convex subsets of topological vector spaces shares this property and that is can be used in lieu of compactness in a variety of cases. Specifically, we establish convex compactness for certain familiar classes of subsets of the set of positive random variables under the topology induced by convergence in probability. Two applications in infinite-dimensional optimization—attainment of infima and a version of the Minimax theorem—are given. Moreover, a new fixed-point theorem of the Knaster-Kuratowski-Mazurkiewicz-type is derived and used to prove a general version of the Walrasian excess-demand theorem.     

Relations among Henstock,McShane and Pettis integrals for multifunctions with compact convex values

Fremlin (Ill J Math 38:471–479, 1994) proved that a Banach space valued function is McShane integrable if and only if it is Henstock and Pettis integrable. In this paper we prove that the result remains valid also in case of multifunctions with compact convex values being subsets of an arbitrary Banach space (see Theorem 3.4). Di Piazza and Musia? (Monatsh Math 148:119–126, 2006) proved that if $X$ is a separable Banach space, then each Henstock integrable multifunction which takes as its values convex compact subsets of $X$ is a sum of a McShane integrable multifunction and a Henstock integrable function. Here we show that such a decomposition is true also in case of an arbitrary Banach space (see Theorem 3.3). We prove also that Henstock and McShane integrable multifunctions possess Henstock and McShane (respectively) integrable selections (see Theorem 3.1).     

Minimizing Irregular Convex Functions: Ulam Stability for Approximate Minima

The main concern of this article is to study Ulam stability of the set of ε-approximate minima of a proper lower semicontinuous convex function bounded below on a real normed space X, when the objective function is subjected to small perturbations (in the sense of Attouch & Wets). More precisely, we characterize the class all proper lower semicontinuous convex functions bounded below such that the set-valued application which assigns to each function the set of its ε-approximate minima is Hausdorff upper semi-continuous for the Attouch–Wets topology when the set $\mathcal{C}(X)$ of all the closed and nonempty convex subsets of X is equipped with the Hausdorff topology. We prove that a proper lower semicontinuous convex function bounded below has Ulam-stable ε-approximate minima if and only if the boundary of any of its sublevel sets is bounded.     

Iterative Regularization Methods for the Multiple-Sets Split Feasibility Problem in Hilbert Spaces

In this paper, we introduce iterative regularization methods for solving the multiple-sets split feasibility problem, that is to find a point closest to a family of closed convex subsets in one space such that its image under a bounded linear mapping will be closest to another family of closed convex subsets in the image space. We consider the cases, when the families are either finite or infinite. We also give two numerical examples for illustrating our main method.

     

Reduction of quasidifferentials and minimal representations

Some criterias for the non-minimality of pairs of compact convex sets of a real locally convex topological vector space are proved, based on a reduction technique via cutting planes and excision of compact convex subsets. Following an example of J. Grzybowski, we construct a class of equivalent minimal pairs of compact convex sets which are not connected by translations.Corresponding author.     

At infinity of finite-dimensional CAT(0) spaces

We show that any filtering family of closed convex subsets of a finite-dimensional CAT(0) space X has a non-empty intersection in the visual bordification ${ \overline{X} = X \cup \partial X}We show that any filtering family of closed convex subsets of a finite-dimensional CAT(0) space X has a non-empty intersection in the visual bordification [`(X)] = X è?X{ \overline{X} = X \cup \partial X} . Using this fact, several results known for proper CAT(0) spaces may be extended to finite-dimensional spaces, including the existence of canonical fixed points at infinity for parabolic isometries, algebraic and geometric restrictions on amenable group actions, and geometric superrigidity for non-elementary actions of irreducible uniform lattices in products of locally compact groups.     

Remarks on Minimal Sets for Cyclic Mappings in Uniformly Convex Banach Spaces

In this article, we prove that every nonempty and convex pair of subsets of uniformly convex in every direction Banach spaces has the proximal normal structure and then we present a best proximity point theorem for cyclic relatively nonexpansive mappings in such spaces. We also study the structure of minimal sets of cyclic relatively nonexpansive mappings and obtain the existence results of best proximity points for cyclic mappings using some new geometric notions on minimal sets. Finally, we prove a best proximity point theorem for a new class of cyclic contraction-type mappings in the setting of uniformly convex Banach spaces and so, we improve the main conclusions of Eldred and Veeramani.     

On information functions and their polars

LetP be a closed convex cone. Information functions, i.e., nonnegative functions onP which are positively homogeneous and concave, are shown to be in a one-to-one correspondence with certain convex subsets ofP. Information functions are always isotone with respect to the vector ordering induced byP, and this order-preserving property distinguishes them from their convex analogues, gauge functions. A polarity concept for information functions is proposed which slightly deviates from the well-known polarity correspondence for gauge functions. Finally, those functions are characterized which differ from information functions only by some nondecreasing concave transformation.     

The Limiting Normal Cone to Pointwise Defined Sets in Lebesgue Spaces

We consider subsets of Lebesgue spaces which are defined by pointwise constraints. We provide formulas for corresponding variational objects (tangent and normal cones). Our main result shows that the limiting normal cone is always dense in the Clarke normal cone and contains the convex hull of the pointwise limiting normal cone. A crucial assumption for this result is that the underlying measure is non-atomic, and this is satisfied in many important applications (Lebesgue measure on subsets of \(\mathbb {R}^{d}\) or the surface measure on hypersurfaces in \(\mathbb {R}^{d}\)). Finally, we apply our findings to an optimization problem with complementarity constraints in Lebesgue spaces.