Chebyshev sets in hyperspaces over a Minkowski space

Bogdewicz, Agnieszka; Dawson, Robert J. MacG.; Moszyńska, Maria

doi:10.3336/gm.42.1.04

Cited by 2 publications

(2 citation statements)

References 7 publications

(6 reference statements)

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“…Observation 2.3. These results hold also in Z B whenever the unit ball B is strictly convex; Theorem 2.5 of [3] takes the place of Theorem 3.3 of [6].…”

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confidence: 65%

“…In [3], various results from [4] and [6] were generalized to hyperspaces K B and O B over Minkowski spaces with unit ball B. Some classes of Čebyšev set, such as singletons {A} and strongly nested sets, carried over in all cases; others, such as convex sets of singletons in K B and families of translates in O B , did so if and only if B was strictly convex.…”

Section: General Minkowski Spaces and Sim-invariant čEbyšev Setsmentioning

confidence: 99%

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Some Čebyšev sets with dimension d+1 in hyperspaces over R^d

Dawson

2009

Convex and Fractal Geometry

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Abstract. A Čebyšev set in a metric space is one such that every point of the space has a unique nearest neighbour in the set. In Euclidean spaces, this property is equivalent to being closed, convex, and nonempty, but in other spaces classification of Čebyšev sets may be significantly more difficult. In particular, in hyperspaces over normed linear spaces several quite different classes of Čebyšev sets are known, with no unifying description. Some new families of Čebyšev sets in hyperspaces are exhibited, with dimension d + 1 (where d is the dimension of the underlying space). They are constructed as translational closures of appropriate nested arcs.1. Introduction. For any metric space (X, ρ), we define a set A ⊂ X to be a Čebyšev set ("be Čebyšev" or "have the Čebyšev property") if for every x ∈ X there is a unique nearest point in A. This property has been studied extensively for normed linear spaces. For such spaces, the Čebyšev property is related to convexity. For Minkowski spaces (finite-dimensional Banach spaces), every Čebyšev set is convex if balls are smooth, while if the balls are strictly convex, every nonempty closed convex set is Čebyšev [10]. In particular, in Euclidean spaces, the Čebyšev sets are precisely those that are nonempty, closed and convex.As a generalization of the Čebyšev property, we define A to be Čebyšev relative to X 0 if every point in X 0 has a unique nearest point in A [6]. This generalizes the concept of "reach": the reach of a set A can be defined

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“…Observation 2.3. These results hold also in Z B whenever the unit ball B is strictly convex; Theorem 2.5 of [3] takes the place of Theorem 3.3 of [6].…”

mentioning

confidence: 65%

Section: General Minkowski Spaces and Sim-invariant čEbyšev Setsmentioning

confidence: 99%

Some Čebyšev sets with dimension d+1 in hyperspaces over R^d

Dawson

2009

Convex and Fractal Geometry

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Monotone and Čebyšev arcs in hyperspaces

Dawson

2010

J. Geom.

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A set in a metric space is called aČebyšev set if it contains a unique "nearest neighbour" to each point of the space. In this paper we introduce the concept of a monotone arc of convex sets and show that compact monotone arcs have theČebyšev property in the hyperspace of compact strictly convex sets. In the hyperspace of compact convex sets only certain monotone arcs areČebyšev ; these are characterized. Results are also obtained for affine segments and for noncompact monotone arcs.Mathematics Subject Classification (2010). 41A52, 52A20.

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Chebyshev sets in hyperspaces over a Minkowski space

Cited by 2 publications

References 7 publications

Some Čebyšev sets with dimension d+1 in hyperspaces over R^d

Some Čebyšev sets with dimension d+1 in hyperspaces over R^d

Monotone and Čebyšev arcs in hyperspaces

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Chebyshev sets in hyperspaces over a Minkowski space

Cited by 2 publications

References 7 publications

Some Čebyšev sets with dimension d+1 in hyperspaces over Rd

Some Čebyšev sets with dimension d+1 in hyperspaces over Rd

Monotone and Čebyšev arcs in hyperspaces

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Some Čebyšev sets with dimension d+1 in hyperspaces over R^d

Some Čebyšev sets with dimension d+1 in hyperspaces over R^d