This generalizes the concept of “reach”: the reach of a set A can be defined sup are Cebysev in Od. Several of the known results come up as corollaries. In the process, we obtain conditions on the endowed hyperspace topology under which the chaotic behaviour of the map on the base space is inherited by the induced map on the hyperspace. As a generalization of the ˇ Cebyˇ sev property, we define A to be ˇ Cebyˇ sev relative to X 0 if every point in X 0 has a unique nearest point in A. In this paper we wish to relate the dynamics of the base map to the dynamics of the induced map. In particular, in Euclidean spaces, the ˇ Cebyˇ sev sets are precisely those that are nonempty, closed and convex. For Minkowski spaces (finite dimensional Banach spaces), every ˇ Cebyˇ sev set is convex if balls are smooth, while if the balls are strictly convex, every nonempty closed convex set is ˇ Cebyˇ sev. For such spaces, the ˇ Cebyˇ sev property is related to convexity. This property has been studied extensively for normed linear spaces. For any metric space ( X, ρ ), we define a set A ⊂ X to be a ˇ Cebyˇ sev set (“be ˇ Cebyˇ sev” or “have the ˇ Cebyˇ sev property”) if for every x ∈ X there is a unique nearest point in A. They are constructed as translational closures of appropriate nested arcs. Some new families of ˇ Cebyˇ sev sets in hyperspaces are exhibited, with dimension d + 1 (where d is the dimension of the underlying space). In particular, in hyperspaces over normed linear spaces several quite different classes of ˇ Cebyˇ sev sets are known, with no unifying description. In Euclidean spaces, this property is equivalent to being closed, convex, and nonempty, but in other spaces classification of ˇ Cebyˇ sev sets may be sig- nificantly more difficult. A ˇ Cebyˇ sev set in a metric space is one such that every point of the space has a unique nearest neighbour in the set. DAWSON Department of Mathematics and Computing Science, Saint Mary’s University, Halifax, Nova Scotia, B3H 3C3, Canada E-mail: Abstract. TI - Dimension of convex hyperspaces : nonmetric case JO - Compositio Mathematica PY - 1983 PB - Martinus Nijhoff Publishers VL. Zbl0557.**************************************** BANACH CENTER PUBLICATIONS, VOLUME ** INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 200* SOME ˇ CEBY ˇ SEV SETS WITH DIMENSION D +1 IN HYPERSPACES OVER R D ROBERT J. vector spaces of dimension > 1, turning the standard hull operation uniformly. Introduction It was shown in 12, 2. In 57, a study is made of uniform convex hyperspaces, the paper. This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which. Van De Vel: Dimension of convex hyperspaces. some invariants of convex hyperspaces and we derive two combinatorial results on subcontinua in a tree. Van De Vel: On the rank of a topological convexity. Van De Vel: A selection theorem for topological convex structures, to appear. Van De Vel: Finite dimensional convex structures II: the invariants. Van De Vel: Finite dimensional convex structures I: general results. Van De Vel: Pseudo-boundaries and pseudo-interiors for topological convexities. Dimension of Convex Hyperspaces Fundamenta Mathematicae. High-Dimension Multilabel Problems: Convex or Nonconvex Relaxation SIAM Journal on Imaging Sciences - United States doi 10.1137/120900307. Van De Vel: Equality of the Lebesgue and the inductive dimension functions for compact spaces with a uniform convexity. The Dimension of the Convex Kernel and Points of Local Nonconvexity Proceedings of the American Mathematical Society - United States doi 10.1090/s0002-9939-1972-0298549-0. We see that several infinite-dimensional properties are preserved by taking the hyperspace of subsets with at most n points. Van De Vel: Subbases, convex sets, and hyperspaces. We consider infinite-dimensional properties in coarse geometry for hyperspaces consisting of finite subsets of metric spaces with the Hausdorff metric. Lawson: The relation of breadth and co-dimension in topological semilattices II. In this paper we study the topological structure of certain hyperspaces of convex subsets of constant width, equipped with the Hausdorff metric topology. Lawson: The relation of breadth and co-dimension in topological semilattices. Cantor Set Approximations and Dimension Computations in Hyperspaces. Dissertation, University of Washington, Seattle, Washington, 1974. Eckhoff: Der Satz von Radon in konvexen Produktstrukturen II.
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