Convergence in distribution of random metric measure spaces (Λ-coalescent measure trees)

We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all finite subspaces sampled from these spaces converge. This topology is metrized following Gromov’s idea of embedding two metric spaces isometrically into a common metric space combined with the Prohorov metric between probability measures on a fixed metric space. We show that for this topology convergence in distribution follows—provided the sequence is tight—from convergence of all randomly sampled finite subspaces. We give a characterization of tightness based on quantities which are reasonably easy to calculate. Subspaces of particular interest are the space of real trees and of ultra-metric spaces equipped with a probability measure. As an example we characterize convergence in distribution for the (ultra-)metric measure spaces given by the random genealogies of the Λ-coalescents. We show that the Λ-coalescent defines an infinite (random) metric measure space if and only if the so-called “dust-free”-property holds.     

Banach operator pairs and common fixed points in hyperconvex metric spaces

The purpose of this paper is to establish DeMarr’s well-known theorem for an arbitrary family of symmetric Banach operator pairs in hyperconvex metric spaces without the compactness assumption. We also give necessary and sufficient criteria for the existence of a common fixed point of a semigroup of isometric mappings. As an application, several results on the invariant best approximation are proved.     

Saddle surfaces in singular spaces

The notion of a saddle surface is well known in Euclidean space. In this work we extend the idea of a saddle surface to geodesically connected metric spaces. We prove that any solution of the Dirichlet problem for the Sobolev energy in a nonpositively curved space is a saddle surface. Further, we show that the space of saddle surfaces in a nonpositively curved space is a complete space in the Fréchet distance. We also prove a compactness theorem for saddle surfaces in spaces of curvature bounded from above; in spaces of constant curvature we obtain a stronger result based on an isoperimetric inequality for a saddle surface. These results generalize difficult theorems of S.Z. Shefel' on compactness of saddle surfaces in a Euclidean space.

     

Between compactness and completeness

Call a sequence in a metric space cofinally Cauchy if for each positive ε there exists a cofinal (rather than residual) set of indices whose corresponding terms are ε-close. We give a number of new characterizations of metric spaces for which each cofinally Cauchy sequence has a cluster point. For example, a space has such a metric if and only each continuous function defined on it is uniformly locally bounded. A number of results exploit a measure of local compactness functional that we introduce. We conclude with a short proof of Romaguera's Theorem: a metrizable space admits such a metric if and only if its set of points having a compact neighborhood has compact complement.     

Uniform convergence on spaces of multifunctions

We study spaces of multifunctions with closed values, multifunctions with closed graphs, USCO multifunctions, minimal USCO multifunctions and the space of densely continuous forms as metric spaces, equipped with the topology of uniform convergence. We give conditions under which these metric spaces are complete.      

A differential operator and weak topology for Lipschitz maps

We show that the Scott topology induces a topology for real-valued Lipschitz maps on Banach spaces which we call the L-topology. It is the weakest topology with respect to which the L-derivative operator, as a second order functional which maps the space of Lipschitz functions into the function space of non-empty weak^∗ compact and convex valued maps equipped with the Scott topology, is continuous. For finite dimensional Euclidean spaces, where the L-derivative and the Clarke gradient coincide, we provide a simple characterization of the basic open subsets of the L-topology. We use this to verify that the L-topology is strictly coarser than the well-known Lipschitz norm topology. A complete metric on Lipschitz maps is constructed that is induced by the Hausdorff distance, providing a topology that is strictly finer than the L-topology but strictly coarser than the Lipschitz norm topology. We then develop a fundamental theorem of calculus of second order in finite dimensions showing that the continuous integral operator from the continuous Scott domain of non-empty convex and compact valued functions to the continuous Scott domain of ties is inverse to the continuous operator induced by the L-derivative. We finally show that in dimension one the L-derivative operator is a computable functional.     

Fuzzy metric spaces

In this paper, fuzzy metric spaces are redefined, different from the previous ones in the way that fuzzy scalars instead of fuzzy numbers or real numbers are used to define fuzzy metric. It is proved that every ordinary metric space can induce a fuzzy metric space that is complete whenever the original one does. We also prove that the fuzzy topology induced by fuzzy metric spaces defined in this paper is consistent with the given one. The results provide some foundations for the research on fuzzy optimization and pattern recognition.     

The Vietoris hyperspace structure for approach spaces

We show that there exists a natural approach version of the topological Vietoris hyperspace construction [16], [17] which has several advantages over the topological version. In the first place the important structural fact that the Vietoris construction can now also be considered, not only for topological but also intrinsically for metric spaces, but equally important in the second place the fact that we can considerably strengthen fundamental classic results. In this paper we mainly pay attention to properties concerning or involving compactness. As main results, in the first place we prove that it is not merely compactness of the Vietoris hyperspace which is equivalent to compactness of the original space [3] but that actually in the approach setting the indices of compactness [7], [8], [9], [10] numerically completely coincide. In the second place the well-known result [3], [4], [15] which says that if the original space is compact metric then the Vietoris topology is metrizable by the Hausdorff metric gets strengthened in the sense that in the approach setting under the same conditions the Vietoris approach structure actually coincides with the Hausdorff metric. Classic results follow as easy corollaries. Besides these main results we also draw attention to the good functorial relationship between the Vietoris approach structures and the associated topologies.     

On maximal element theorems,variants of Ekeland’s variational principle and their applications

In this paper, we establish several different versions of generalized Ekeland’s variational principle and maximal element theorem for τ$\tau$-functions in ?$?$ complete metric spaces. The equivalence relations between maximal element theorems, generalized Ekeland’s variational principle, generalized Caristi’s (common) fixed point theorems and nonconvex maximal element theorems for maps are also proved. Moreover, we obtain some applications to a nonconvex minimax theorem, nonconvex vectorial equilibrium theorems and convergence theorems in complete metric spaces.     

Mappings onto metric spaces

We give characterizations of perfect images and open and compact images of spaces that can be mapped onto metrizable spaces by a mapping with fibers having a given property $\text{P}$. We use these characterizations to obtain conditions which imply that such images can be mapped onto a metric space by a mapping with fibers satisfying $\text{P}$. Such a treatment includes the investigation of spaces with a weaker metric topology [2, Ch. 5].     

Closure of the set of classical Riemannian spaces

The basic result of the paper is a theorem asserting that the closure of the set of compact Riemannian spaces in the set of all compact metric spaces with inner metric consists precisely of the set of compact metric spaces with bilaterally bounded curvature in the sense of A. D. Aleksandrov. Here the convergence of a sequence of Riemannian spaces in the topology we consider means its Lipschitz convergence to a limit metric space and the uniform bilateral boundedness of the sectional curvatures of the spaces of the sequence. The results obtained are considered in application to the compactness theorem of M. Gromov.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 21, pp. 43–66, 1989.     

Local Currents in Metric Spaces

Ambrosio and Kirchheim presented a theory of currents with finite mass in complete metric spaces. We develop a variant of the theory that does not rely on a finite mass condition, closely paralleling the classical Federer–Fleming theory. If the underlying metric space is an open subset of a Euclidean space, we obtain a natural chain monomorphism from general metric currents to general classical currents whose image contains the locally flat chains and which restricts to an isomorphism for locally normal currents. We give a detailed exposition of the slicing theory for locally normal currents with respect to locally Lipschitz maps, including the rectifiable slices theorem, and of the compactness theorem for locally integral currents in locally compact metric spaces, assuming only standard results from analysis and measure theory.     

The Mazurkiewicz Distance and Sets that are Finitely Connected at the Boundary

We study local connectedness, local accessibility and finite connectedness at the boundary, in relation to the compactness of the Mazurkiewicz completion of a bounded domain in a metric space. For countably connected planar domains we obtain a complete characterization. It is also shown exactly which parts of this characterization fail in higher dimensions and in metric spaces.     

Existence of vector mixed variational inequalities in Banach spaces

The purpose of this paper is to study the solvability for vector mixed variational inequalities (for short, VMVI) in Banach spaces. Utilizing Ky Fan’s Lemma and Nadler’s theorem, we derive the solvability for VMVIs with compositely monotone vector multifunctions. On the other hand, we first introduce the concepts of compositely complete semicontinuity and compositely strong semicontinuity for vector multifunctions. Then we prove the solvability for VMVIs without monotonicity assumption by using these concepts and by applying Brouwer’s fixed point theorem. The results presented in this paper are extensions and improvements of some earlier and recent results in the literature.     

Compactness for manifolds and integral currents with bounded diameter and volume

By Gromov??s compactness theorem for metric spaces, every uniformly compact sequence of metric spaces admits an isometric embedding into a common compact metric space in which a subsequence converges with respect to the Hausdorff distance. Working in the class of oriented k-dimensional Riemannian manifolds (with boundary) and, more generally, integral currents in metric spaces in the sense of Ambrosio?CKirchheim and replacing the Hausdorff distance with the filling volume or flat distance, we prove an analogous compactness theorem in which however we only assume uniform bounds on volume and diameter.     

Equilibrium maps between metric spaces

We show the existence of harmonic mappings with values in possibly singular and not necessarily locally compact complete metric length spaces of nonpositive curvature in the sense of Alexandrov. As a technical tool, we show that any bounded sequence in such a space has a subsequence whose mean values converge. We also give a general definition of harmonic maps between metric spaces based on mean value properties and-convergence.     

Quasi-metrics and monotone normality

The purpose of this paper is to study which quasi-metrizable spaces are monotonically normal. In particular, we provide a sufficient condition for a quasi-metrizable space to be monotonically normal. This enables us to prove the monotone normality of a certain amount of interesting examples of quasi-metric spaces; for instance, we show that the continuous poset of formal balls of a metric space, endowed with the Scott topology, is a monotonically normal quasi-metrizable space.     

A Kuratowski–Mrówka theorem in approach theory

In this paper we give a number of arguments why, in approach theory, the notion of compactness which from the intrinsic categorical point of view seems most satisfying is 0-compactness, i.e., measure of compactness equal to zero. It was already known from [R. Lowen, Kuratowski's measure of noncompactness revisited, Quart. J. Math. Oxford 39 (1988) 235–254] that measure of compactness has good properties and good interpretations for both topological and metric approach spaces. Here, introducing notions of closed and proper mappings in approach theory, which satisfy all the intrinsic categorical axioms put forth in [Clementino et al., A functional approach to topology, in: M.C. Pedicchio, W. Tholen (Eds.) Categorical Foundations Special Topics in Order, Topology, Algebra, and Sheaf Theory, Cambridge University Press, 2003], we prove fundamental results concerning these concepts, also linked to 0-compactness, and we give a Kuratowski–Mrówka-type characterization of 0-compactness.     

The structure of singular spaces of dimension 2

In this paper, we study the structure of locally compact metric spaces of Hausdorff dimension 2. If such a space has non-positive curvautre and a local cone structure, then every simple closed curve bounds a conformal disk. On a surface (a topological manifold of dimension 2), a distance function with non-positive curvature and whose metric topology is equivalent to the surface topology gives a structure of a Riemann surface. The construction of conformal disks in these spaces uses minimal surface theory; in particular, the solution of the Plateau Problem in metric spaces of non-positive curvature. Received: 18 November 1997/ Revised versions: 15 January and 7 June 1999     

On weak compactness and countable weak compactness in fixed point theory

We prove that weak compactness and countable weak compactness in metric spaces are not equivalent. However, if the metric space has normal structure, they are equivalent. It follows that some fixed point theorems proved recently are consequences of a classical theorem of Kirk.