Optimal position of compact sets and the Steiner problem in spaces with Euclidean Gromov-Hausdorff metric

2020 ◽  
Vol 211 (10) ◽  
pp. 1382-1398
Author(s):  
O. S. Malysheva
Keyword(s):  
Hausdorff Metric ◽  
Steiner Problem ◽  
Compact Sets ◽  
Optimal Position

Čebyšev Sets in Hyperspaces over Rn

2009 ◽  
Vol 61 (2) ◽  
pp. 299-314 ◽  
Author(s):  
Robert J. MacG. Dawson and ◽  
Maria Moszyńska
Keyword(s):  
Metric Space ◽  
Convex Sets ◽  
Compact Convex ◽  
Hausdorff Metric ◽  
Linear Structure ◽  
Convex Compact ◽  
Nearest Neighbour ◽  
Compact Sets ◽  
Strictly Convex ◽  
Convex Compact Sets

Abstract. A set in a metric space is called a Čebyšev set if it has a unique “nearest neighbour” to each point of the space. In this paper we generalize this notion, defining a set to be Čebyšev relative to another set if every point in the second set has a unique “nearest neighbour” in the first. We are interested in Čebyšev sets in some hyperspaces over Rn, endowed with the Hausdorff metric, mainly the hyperspaces of compact sets, compact convex sets, and strictly convex compact sets. We present some new classes of Čebyšev and relatively Čebyšev sets in various hyperspaces. In particular, we show that certain nested families of sets are Čebyšev. As these families are characterized purely in terms of containment,without reference to the semi-linear structure of the underlyingmetric space, their properties differ markedly from those of known Čebyšev sets.

Fermat-Steiner problem in the space of compact subsets of [IMG align=ABSMIDDLE alt=$ \mathbb R^m$]tex_sm_4905_img1[/IMG], endowed with the Hausdorff metric

2021 ◽  
Vol 212 (1) ◽  
Author(s):  
Arsen Khachaturovich Galstyan ◽  
Alexandr Olegovich Ivanov ◽  
Alexey Avgustinovich Tuzhilin
Keyword(s):  
Hausdorff Metric ◽  
Steiner Problem ◽  
Compact Subsets

scholarly journals Fermat–Steiner problem in the metric space of compact sets endowed with Hausdorff distance

2016 ◽  
Vol 108 (2) ◽  
pp. 575-590
Author(s):  
Alexander Ivanov ◽  
Alexander Tropin ◽  
Alexey Tuzhilin
Keyword(s):  
Metric Space ◽  
Hausdorff Distance ◽  
Steiner Problem ◽  
Compact Sets

CONVERGENCE OF JULIA SETS IN THE APPROXIMATION OF λez BY λ[1+(z/d)]d

1993 ◽  
Vol 03 (02) ◽  
pp. 257-270 ◽  
Author(s):  
BERND KRAUSKOPF
Keyword(s):  
Periodic Orbit ◽  
Julia Set ◽  
Julia Sets ◽  
Hausdorff Metric ◽  
Cantor Set ◽  
Compact Sets

The polynomials Pd,λ(z)≔λ[1+(z/d)]dconverge uniformly on compact sets to Eλ(z)≔λez. What this convergence means for the dynamics of these functions when iterated was first studied in Devaney et al. [preprint]. Here we show the convergence of the corresponding Julia sets in the Hausdorff metric for two cases: (1) for λ such that Eλ has an attracting periodic orbit, in which case its Julia set is a Cantor set of curves, and (2) for λ such that the Julia set of Eλ is the whole plane [Formula: see text]. Finally, we give the key ideas of the algorithms designed to illustrate this convergence.

scholarly journals A note on compact sets in spaces of subsets

1988 ◽  
Vol 38 (3) ◽  
pp. 393-395 ◽  
Author(s):  
Phil Diamond ◽  
Peter Kloeden
Keyword(s):  
Metric Space ◽  
Complete Metric Space ◽  
Hausdorff Metric ◽  
Compact Sets ◽  
Selection Theorem ◽  
Starshaped Sets ◽  
Nonempty Compact ◽  
Compact Subsets

A simple characterisation is given of compact sets of the space K(X), of nonempty compact subsets of a complete metric space X, with the Hausdorff metric dH. It is used to give a new proof of the Blaschke selection theorem for compact starshaped sets.

scholarly journals On Statistical Properties of Sets Fulfilling Rolling-Type Conditions

2012 ◽  
Vol 44 (2) ◽  
pp. 311-329 ◽  
Author(s):  
A. Cuevas ◽  
R. Fraiman ◽  
B. Pateiro-López
Keyword(s):  
Convex Hull ◽  
Level Set ◽  
Hausdorff Metric ◽  
Two Dimensional ◽  
Compact Sets ◽  
Rolling Condition ◽  
Set Estimation ◽  
Estimation Problems ◽  
Target Set ◽  
Uniformly Bounded

Motivated by set estimation problems, we consider three closely related shape conditions for compact sets: positive reach, r-convexity, and the rolling condition. First, the relations between these shape conditions are analyzed. Second, for the estimation of sets fulfilling a rolling condition, we obtain a result of ‘full consistency’ (i.e. consistency with respect to the Hausdorff metric for the target set and for its boundary). Third, the class of uniformly bounded compact sets whose reach is not smaller than a given constant r is shown to be a P-uniformity class (in Billingsley and Topsøe's (1967) sense) and, in particular, a Glivenko-Cantelli class. Fourth, under broad conditions, the r-convex hull of the sample is proved to be a fully consistent estimator of an r-convex support in the two-dimensional case. Moreover, its boundary length is shown to converge (almost surely) to that of the underlying support. Fifth, the above results are applied to obtain new consistency statements for level set estimators based on the excess mass methodology (see Polonik (1995)).

scholarly journals On Statistical Properties of Sets Fulfilling Rolling-Type Conditions

2012 ◽  
Vol 44 (02) ◽  
pp. 311-329 ◽  
Author(s):  
A. Cuevas ◽  
R. Fraiman ◽  
B. Pateiro-López
Keyword(s):  
Convex Hull ◽  
Level Set ◽  
Hausdorff Metric ◽  
Two Dimensional ◽  
Compact Sets ◽  
Rolling Condition ◽  
Set Estimation ◽  
Estimation Problems ◽  
Target Set ◽  
Uniformly Bounded

Motivated by set estimation problems, we consider three closely related shape conditions for compact sets: positive reach,r-convexity, and the rolling condition. First, the relations between these shape conditions are analyzed. Second, for the estimation of sets fulfilling a rolling condition, we obtain a result of ‘full consistency’ (i.e. consistency with respect to the Hausdorff metric for the target set and for its boundary). Third, the class of uniformly bounded compact sets whose reach is not smaller than a given constantris shown to be aP-uniformity class (in Billingsley and Topsøe's (1967) sense) and, in particular, a Glivenko-Cantelli class. Fourth, under broad conditions, ther-convex hull of the sample is proved to be a fully consistent estimator of anr-convex support in the two-dimensional case. Moreover, its boundary length is shown to converge (almost surely) to that of the underlying support. Fifth, the above results are applied to obtain new consistency statements for level set estimators based on the excess mass methodology (see Polonik (1995)).

DEFORMATION OF F-TILINGSVERSUSDEFORMATION OF ISOMETRIC FOLDINGS

2012 ◽  
Vol 23 (09) ◽  
pp. 1250092 ◽  
Author(s):  
CATARINA P. AVELINO ◽  
ALTINO F. SANTOS
Keyword(s):  
Hausdorff Metric ◽  
Compact Sets ◽  
Continuous Deformation ◽  
Natural Way

We present some relations between deformation of spherical isometric foldings and deformation of spherical f-tilings. The natural way to deform f-tilings is based on the Hausdorff metric on compact sets. It is conjectured that any f-tiling is (continuously) deformable in the standard f-tiling τs= {(x, y, z) ∈ S2: z = 0} and it is shown that the deformation of f-tilings does not induce a continuous deformation on its associated isometric foldings.

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