New metric properties for prox-regular sets

Mathematical Programming - In this paper, we present diverse new metric properties that prox-regular sets shared with convex ones. At the heart of our work lie the Legendre-Fenchel transform and...     

Symmetry properties of resolving sets and metric bases in hypercubes

In this paper we consider some special characteristics of distances between vertices in the \(n\)-dimensional hypercube graph \(Q_n\) and, as a consequence, the corresponding symmetry properties of its resolving sets. It is illustrated how these properties can be implemented within a simple greedy heuristic in order to find efficiently an upper bound of the so called metric dimension \(\beta (Q_n)\) of \(Q_n\), i.e. the minimal cardinality of a resolving set in \(Q_n\). This heuristic was applied to generate upper bounds of \(\beta (Q_n)\) for \(n\) up to \(22\), which are for \(n\ge 19\) better than the existing ones. Starting from these new bounds, some existing upper bounds for \(23\le n\le 90\) are improved by a dynamic programming procedure.     

On the variational convergence of fuzzy sets in metric spaces

Kaleva [9] has studied the relationships between the metric convergencesH andD of fuzzy convex sets on Euclidean spaces. The distanceH between two fuzzy set is given by Hausdorff distance of their sendographs, whileD is the supremum of the Hausdorff distances of the level sets corresponding to the fuzzy sets. The aim of this paper is to compareH andD with the variational convergence, called γ-convergence (see De Giorgi and Franzoni [3]). Our analysis which is carried out in the setting of metric spaces (not necessarily locally compact or vector spaces), improves Kaleva's results.

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Sunto Kaleva ha investigato in [9] le relazioni esistenti tra due convergenze metriche, detteH eD, di sottoinsiemi fuzzy di spazi euclidei finito-dimensionali. In questo articolo le convergenzeH eD (la loro definizione dipende dalla distanza di Hausdorff tra insiemi compatti) sono confrontate con la convergenza variazionale, detta γ-convergenza, introdotta da De Giorgi and Franzoni in [3] nel contesto degli spazi topologici. Tale confronto con la γ-convergenza (vedi Teorema 3.7), svolto nell'ambito degli spazi metrici (non necessariamente, localmente compatti o lineari) migliora ed estende i precedenti risultati di Kaleva.

     

On a metric for the class of compact convex sets

A distance function, defined in [12], for the class of compact convex sets inn-space is introduced in a new way, and some of its properties are developed. This concept is compared with some traditional distance functions for convex sets.     

Rank-determining sets of metric graphs

A metric graph is a geometric realization of a finite graph by identifying each edge with a real interval. A divisor on a metric graph Γ is an element of the free abelian group on Γ. The rank of a divisor on a metric graph is a concept appearing in the Riemann-Roch theorem for metric graphs (or tropical curves) due to Gathmann and Kerber, and Mikhalkin and Zharkov. We define a rank-determining set of a metric graph Γ to be a subset A of Γ such that the rank of a divisor D on Γ is always equal to the rank of D restricted on A. We show constructively in this paper that there exist finite rank-determining sets. In addition, we investigate the properties of rank-determining sets in general and formulate a criterion for rank-determining sets. Our analysis is based on an algorithm to derive the v₀-reduced divisor from any effective divisor in the same linear system.     

Continuity of the metric projection,structural and approximate properties of sets

Translated from Matematicheskie Zametki, Vol. 47, No. 2, pp. 137–148, February, 1990.     

Supremum metric and relatively compact sets of fuzzy sets

The supremum metric D between fuzzy subsets of a metric space is the supremum of the Hausdorff distances of the corresponding level sets. In this paper some new criteria of compactness with respect to the distance D are given; they concern arbitrary fuzzy sets (see Theorem 7), fuzzy sets having no proper local maximum points (see Theorem 12) and, finally, fuzzy sets with convex sendograph (see Theorem 13). In order to compare results with a previous characterization of compactness of Diamond–Kloeden, the criteria will be expressed by equi-(left/right)-continuity. In the proofs a first author's purely topological criterion of D  -compactness and a variational convergence (called Γ$\Gamma$-convergence) which was introduced by De Giorgi and Franzoni, are fundamental.     

Gated sets in metric spaces

The concept of a gated subset in a metric space is studied, and it is shown that properties of disjoint pairs of gated subsets can be used to investigate projections in Tits buildings.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

Polar sets on metric spaces

We show that if is a proper metric measure space equipped with a doubling measure supporting a Poincaré inequality, then subsets of with zero -capacity are precisely the -polar sets; that is, a relatively compact subset of a domain in is of zero -capacity if and only if there exists a -superharmonic function whose set of singularities contains the given set. In addition, we prove that if is a -hyperbolic metric space, then the -superharmonic function can be required to be -superharmonic on the entire space . We also study the the following question: If a set is of zero -capacity, does there exist a -superharmonic function whose set of singularities is precisely the given set?

     

On metric properties of certain clique graphs

The paper provides a unified point of view on some classes of graphs: clique graphs, weakly geodetic graphs, ptolemaic graphs and Husimi trees. A purely metric characterization of Husimi trees is given.     

On some variational properties of metric spaces

We use the basic formulation of Ekeland’s variational principle to establish characterizations of complete path metric spaces which, being described in terms of the strong slope, are called coherent as in [3]. We also provide some basic nonlinear error bound and metric regularity results, in the context of coherent spaces.     

On arithmetic properties of Cantor sets

Science China Mathematics - In this paper, we study three types of Cantor sets. For any integer m ? 4, we show that every real number in [0, k] is the sum of at most k m-th powers of...     

Self-similar sets in complete metric spaces

We develop a theory for Hausdorff dimension and measure of self-similar sets in complete metric spaces. This theory differs significantly from the well-known one for Euclidean spaces. The open set condition no longer implies equality of Hausdorff and similarity dimension of self-similar sets and that has nonzero Hausdorff measure in this dimension. We investigate the relationship between such properties in the general case.

     

Large null sets in metric spaces

It is shown that every locally compact σ-compact metric space endowed with a Borel measure related to the metric by a natural condition contains sets of measure zero which are extremely large in the sense of cardinality, Hausdorff dimension and Baire category classification.