Chebyshev centres, Jung constants, and their applications

Алимов, А. Р.; Царьков, Игорь Германович

doi:10.1070/rm9839

Cited by 21 publications

(8 citation statements)

References 124 publications

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“…It is also proved in [19] that all C 2 -smooth convex curves have good approximations by polygonal chains in the sense of the self Chebyshev radius (1). This observation leads to natural extremal problems for convex polygons.…”

Section: Introduction and The Main Resultsmentioning

confidence: 92%

“…For a given point A ∈ Γ we have d(A, A i ) = d(A, A j ) if and only if A is on the straight line trough the point 1 2 (A i + A j ) perpendicular to [A i , A j ]. Hence, we have only finite numbers of points A ∈ Γ such that the set F (A) has more that one point.…”

Section: For a Given Pointmentioning

confidence: 99%

“…Moreover, this maximum perimeter coincides with the perimeter of the regular 2ngon of diameter d, as E. Makai, J. noticed (a private conversation). Indeed, if P n is such a n-gon, then the diameter of the polygon 1 2 (P n − P n ) (this is the central symmetral of P n , that have the same perimeter as P n has) is also equal to d, 1 2 (P n − P n ) is inscribed into a circle of diameter d, and has ≤ 2n sides. So, this perimeter is no more than the perimeter of the regular 2n-gon incribed into this circle (that can obtained via the above procedure from a regular n-gon of diameter d).…”

Section: Computations and Further Conjecturesmentioning

confidence: 99%

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The extreme polygons for the self Chebyshev radius of the boundary

Nikitenko¹,

Nikonorov²

2023

Preprint

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The paper is devoted to some extremal problems for convex polygons on the Euclidean plane, related to the concept of self Chebyshev radius for the polygon boundary. We consider a general problem of minimization of the perimeter among all n-gons with a fixed self Chebyshev radius of the boundary. The main result of the paper is the complete solution of the mentioned problem for n = 4: We proved that the quadrilateral of minimum perimeter is a so called magic kite, that verified the corresponding conjecture by Rolf Walter.

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Section: Introduction and The Main Resultsmentioning

confidence: 92%

Section: For a Given Pointmentioning

confidence: 99%

Section: Computations and Further Conjecturesmentioning

confidence: 99%

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The extreme polygons for the self Chebyshev radius of the boundary

Nikitenko¹,

Nikonorov²

2023

Preprint

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“…Hence py 1 , y 2 q P cent Y pB, δq. Then there exists y 1 1 P cent Y 1 pB 1 q and y 1 2 P cent Y 2 pB 2 q such that }py 1 , y 2 q ´py 1 1 , y 1 2 q} ď ε. Thus }y 1 ´y1 1 } ď ε.…”

Section: Now Using the Arguments Above We Obtainmentioning

confidence: 99%

On property-$(P_1)$ and semi-continuity properties of restricted Chebyshev-center maps in $\ell_{\infty}$-direct sums

Thomas¹

2023

Preprint

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For a compact Hausdorff space S, we prove that the closed unit ball of a closed linear subalgebra of the space of real-valued continuous functions on S, denoted by CpSq, satisfies property-pP1q (the setvalued generalization of strong proximinality) for the non-empty closed bounded subsets of the bidual of CpSq. Various stability results related to property-pP1q and semi-continuity properties of restricted Chebyshevcenter maps are also established. As a consequence, we derive that if Y is a proximinal finite co-dimensional subspace of c0 then the closed unit ball of Y satisfies property-pP1q for the non-empty closed bounded subsets of ℓ8 and the restricted Chebyshev-center map of the closed unit ball of Y is Hausdorff metric continuous on the class of non-empty closed bounded subsets of ℓ8. We also investigate a variant of the transitivity property, similar to the one discussed in [C. R. Jayanarayanan and T. Paul, Strong proximinality and intersection properties of balls in Banach spaces,

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“…This is evident through [2]- [4], [6], [9], [10], [12], [14] and the references therein. One can also refer [1], which is a recent survey article discussing the current state of few parts of this study.…”

mentioning

confidence: 99%

Some stability properties of restricted Chebyshev centers in Banach spaces

Thomas¹

2021

Preprint

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In this paper, we discuss the stability of (restricted) Chebyshev centers in few function spaces. For an extremally disconnected compact Hausdorff space K and a finite dimensional Banach space X, we prove the existence of Chebyshev centers of closed bounded subsets of the space of X-valued continuous functions on K, CpK, Xq and that of compact subsets of M -ideals in CpK, Xq. It is also proved that the existence of restricted Chebyshev centers is stable in the space of vector-valued bounded functions on an arbitrary set. For a compact Hausdorff space K, we provide an explicit description of the Chebyshev centers of closed bounded subsets of an M -summand in CpK, Rq. The stability of continuity properties of the Chebyshev-center map of a Banach space X in dependence on the continuity properties of the Chebyshev-center map of CpK, Xq is also studied.

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Chebyshev centres, Jung constants, and their applications

Cited by 21 publications

References 124 publications

The extreme polygons for the self Chebyshev radius of the boundary

The extreme polygons for the self Chebyshev radius of the boundary

On property-$(P_1)$ and semi-continuity properties of restricted Chebyshev-center maps in $\ell_{\infty}$-direct sums

Some stability properties of restricted Chebyshev centers in Banach spaces

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