Chebyshev center and inscribed balls: properties and calculations

2021 ◽  
Author(s):  
Maxim V. Balashov
Keyword(s):  
Chebyshev Center

On Asymptotic Centers and Fixed Points of Nonexpansive Mappings

1980 ◽  
Vol 32 (2) ◽  
pp. 421-430 ◽  
Author(s):  
Teck-Cheong Lim
Keyword(s):  
Banach Space ◽  
Fixed Points ◽  
Nonempty Subset ◽  
Nonexpansive Mappings ◽  
Chebyshev Center ◽  
Bounded Subset ◽  
Uniformly Convex ◽  
Weakly Compact ◽  
Asymptotic Centers ◽  
Image Position

Let X be a Banach space and B a bounded subset of X. For each x ∈ X, define R(x) = sup{‖x – y‖ : y ∈ B}. If C is a nonempty subset of X, we call the number R = inƒ{R(x) : x ∈ C} the Chebyshev radius of B in C and the set the Chebyshev center of B in C. It is well known that if C is weakly compact and convex, then and if, in addition, X is uniformly convex, then the Chebyshev center is unique; see e.g., [9].Let {Bα : α ∈ ∧} be a decreasing net of bounded subsets of X. For each x ∈ X and each α ∈ ∧, define

scholarly journals Some remarks on the location of fixed points

1984 ◽  
Vol 37 (3) ◽  
pp. 358-365
Author(s):  
Michael Edelstein ◽  
Daryl Tingley
Keyword(s):  
Banach Space ◽  
Fixed Points ◽  
Convex Subset ◽  
Compact Convex ◽  
Compact Convex Subset ◽  
Chebyshev Center ◽  
Weakly Compact ◽  
Asymptotic Center

AbstractSeveral procedures for locating fixed points of nonexpansive selfmaps of a weakly compact convex subset of a Banach space are presented. Some of the results involve the notion of an asymptotic center or a Chebyshev center.

scholarly journals On Chebyshev centers in metric spaces

2019 ◽  
Vol 106 (120) ◽  
pp. 47-51
Author(s):  
T.D. Narang
Keyword(s):  
Metric Space ◽  
Existence And Uniqueness ◽  
Metric Spaces ◽  
Chebyshev Center ◽  
The Subject

A Chebyshev center of a set A in a metric space (X,d) is a point of X best approximating the set A i.e., it is a point x0 ? X such that supy?A d(x0,y) = infx?X supy?A d(x,y). We discuss the existence and uniqueness of such points in metric spaces thereby generalizing and extending several known result on the subject.

scholarly journals Characterizations of inner product spaces by means of norm one points

2005 ◽  
Vol 97 (1) ◽  
pp. 104
Author(s):  
José Mendoza ◽  
Tijani Pakhrou
Keyword(s):  
Convex Hull ◽  
Unit Sphere ◽  
Product Space ◽  
Normed Linear Space ◽  
Inner Product ◽  
Chebyshev Center ◽  
Product Spaces ◽  
Inner Product Spaces ◽  
Point Subset ◽  
Real Normed Linear Space

Let $X$ be a a real normed linear space of dimension at least three, with unit sphere $S_X$. In this paper we prove that $X$ is an inner product space if and only if every three point subset of $S_X$ has a Chebyshev center in its convex hull. We also give other characterizations expressed in terms of centers of three point subsets of $S_X$ only. We use in these characterizations Chebyshev centers as well as Fermat centers and $p$-centers.

scholarly journals Methods of optimization of Hausdorff distance between convex rotating figures

2020 ◽  
Vol 30 (4) ◽  
pp. 429-442
Author(s):  
Pavel Lebedev ◽  
Vladimir Ushakov
Keyword(s):  
Hausdorff Distance ◽  
Software Package ◽  
Convex Set ◽  
Iterative Algorithms ◽  
The Other ◽  
Compact Set ◽  
Chebyshev Center ◽  
Convex Polygons ◽  
Parallel Transfer ◽  
Differential Properties

We studied the problem of optimizing the Hausdorff distance between two convex polygons. Its minimization is chosen as the criterion of optimality. It is believed that one of the polygons can make arbitrary movements on the plane, including parallel transfer and rotation with the center at any point. The other polygon is considered to be motionless. Iterative algorithms for the phased shift and rotation of the polygon are developed and implemented programmatically, providing a decrease in the Hausdorff distance between it and the fixed polygon. Theorems on the correctness of algorithms for a wide class of cases are proved. Moreover, the geometric properties of the Chebyshev center of a compact set and the differential properties of the Euclidean function of distance to a convex set are essentially used. When implementing the software package, it is possible to run multiple times in order to identify the best found polygon position. A number of examples are simulated.

Conditional Chebyshev center of a bounded set of continuous functions

1975 ◽  
Vol 18 (1) ◽  
pp. 622-627 ◽  
Author(s):  
A. L. Garkavi ◽  
V. N. Zamyatin
Keyword(s):  
Continuous Functions ◽  
Chebyshev Center ◽  
Bounded Set

scholarly journals Some Results on Iterative Proximal Convergence and Chebyshev Center

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Laishram Shanjit ◽  
Yumnam Rohen ◽  
Sumit Chandok ◽  
M. Bina Devi
Keyword(s):  
Banach Space ◽  
Reflexive Banach Space ◽  
Best Proximity Point ◽  
Normal Structure ◽  
Sufficient Condition ◽  
Chebyshev Center ◽  
Opial’S Condition ◽  
Relatively Nonexpansive

In this paper, we prove a sufficient condition that every nonempty closed convex bounded pair M , N in a reflexive Banach space B satisfying Opial’s condition has proximal normal structure. We analyze the relatively nonexpansive self-mapping T on M ∪ N satisfying T M ⊆ M and T N ⊆ N , to show that Ishikawa’s and Halpern’s iteration converges to the best proximity point. Also, we prove that under relatively isometry self-mapping T on M ∪ N satisfying T N ⊆ N and T M ⊆ M , Ishikawa’s iteration converges to the best proximity point in the collection of all Chebyshev centers of N relative to M . Some illustrative examples are provided to support our results.

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