On complete metrizability of the Hausdorff metric topology

We present a natural topology compatible with the Mosco convergence of sequences of closed convex sets in a reflexive space, and characterise the topology in terms of the continuity of the distance between convex sets and fixed weakly compact ones. When the space is separable, the topology is Polish. As an application, we show that in this context, most closed convex sets are almost Chebyshev, a result that fails for the stronger Hausdorff metric topology.

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Paths in hyperspaces

Applied General Topology ◽

10.4995/agt.2003.2040 ◽

2003 ◽

Vol 4 (2) ◽

pp. 377 ◽

Cited By ~ 5

Author(s):

Camillo Constantini ◽

Wieslaw Kubís

Keyword(s):

Metric Space ◽

Metric Spaces ◽

Hausdorff Metric ◽

Absolute Retract ◽

Almost Convex ◽

Convex Metric Spaces ◽

Metric Topology ◽

Hausdorff Metric Topology ◽

Bounded Sets ◽

Separable Metric Space

<p>We prove that the hyperspace of closed bounded sets with the Hausdor_ topology, over an almost convex metric space, is an absolute retract. Dense subspaces of normed linear spaces are examples of, not necessarily connected, almost convex metric spaces. We give some necessary conditions for the path-wise connectedness of the Hausdorff metric topology on closed bounded sets. Finally, we describe properties of a separable metric space, under which its hyperspace with the Wijsman topology is path-wise connected.</p>

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Cofinal completeness of the Hausdorff metric topology

Fundamenta Mathematicae ◽

10.4064/fm208-1-5 ◽

2010 ◽

Vol 208 (1) ◽

pp. 75-85 ◽

Cited By ~ 4

Author(s):

Gerald Beer ◽

Giuseppe Di Maio

Keyword(s):

Hausdorff Metric ◽

Metric Topology ◽

Hausdorff Metric Topology

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Metric Fourier Approximation of Set-Valued Functions of Bounded Variation

Journal of Fourier Analysis and Applications ◽

10.1007/s00041-021-09812-7 ◽

2021 ◽

Vol 27 (2) ◽

Author(s):

Elena E. Berdysheva ◽

Nira Dyn ◽

Elza Farkhi ◽

Alona Mokhov

Keyword(s):

Fourier Series ◽

Bounded Variation ◽

Error Bounds ◽

Metric Spaces ◽

Hausdorff Metric ◽

Dirichlet Kernel ◽

Partial Sums ◽

Functions Of Bounded Variation ◽

Moduli Of Continuity ◽

Fourier Approximation

AbstractWe introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selections of the approximated multifunction at a point of discontinuity. Our error bounds are obtained with the help of the new notions of one-sided local moduli and quasi-moduli of continuity which we discuss more generally for functions with values in metric spaces.

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Hausdorff GAN: Improving GAN Generation Quality With Hausdorff Metric

IEEE Transactions on Cybernetics ◽

10.1109/tcyb.2021.3062396 ◽

2021 ◽

pp. 1-13 ◽

Cited By ~ 1

Author(s):

Wei Li ◽

Zhixuan Liang ◽

Ping Ma ◽

Ruobei Wang ◽

Xiaohui Cui ◽

...

Keyword(s):

Hausdorff Metric

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An Analytical Study in Multi-physics and Multi-criteria Shape Optimization

Journal of Optimization Theory and Applications ◽

10.1007/s10957-021-01841-y ◽

2021 ◽

Author(s):

Hanno Gottschalk ◽

Marco Reese

Keyword(s):

Shape Optimization ◽

Physical System ◽

Pareto Front ◽

Optimization Problem ◽

Analytical Study ◽

Static Pressure ◽

Hausdorff Metric ◽

Objective Functions ◽

Hölder Continuous ◽

Turbine Vane

AbstractA simple multi-physical system for the potential flow of a fluid through a shroud, in which a mechanical component, like a turbine vane, is placed, is modeled mathematically. We then consider a multi-criteria shape optimization problem, where the shape of the component is allowed to vary under a certain set of second-order Hölder continuous differentiable transformations of a baseline shape with boundary of the same continuity class. As objective functions, we consider a simple loss model for the fluid dynamical efficiency and the probability of failure of the component due to repeated application of loads that stem from the fluid’s static pressure. For this multi-physical system, it is shown that, under certain conditions, the Pareto front is maximal in the sense that the Pareto front of the feasible set coincides with the Pareto front of its closure. We also show that the set of all optimal forms with respect to scalarization techniques deforms continuously (in the Hausdorff metric) with respect to preference parameters.

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