Chebyshev centres, Jung constants, and their applications. (English. Russian original) Zbl 1435.41028
Russ. Math. Surv. 74, No. 5, 775-849 (2019); translation from Usp. Mat. Nauk 74, No. 5, 3-82 (2019).
This nice survey article covers various aspectes of one of the most useful and important geometric concept, that of Chebyshev centre of a bounded set. With 169 references, the authors cover, very impressively, the work done in this area, over several decades and across several continents, on this fundamental notion.
This article will become a ‘must read’ for any one interested in these aspects of approximation theory and its applications. The Russian school has originated and pioneered some of the fundamental concepts covered here. It continues to be a very active area. By bringing to the fore a lot of work (both classical and current) done by the European and Indian schools, the authors have done a great service.
For a bounded set \(M\), the Chebyshev centre, is the centre of a ball of smallest radius containing the set. Thus it is a point that best approximates the entire set.
Let \(Z(M)\) denote the set of Chebyshev centres of \(M\) and the set-valued map, \(M \rightarrow Z(M)\) is called the Chebyshev centre map. For a non-empty set \(Y \subset X\), \(Z_Y(M)\) denotes the set of relative Chebyshev centres. This is a convex set, when \(Y\) is.
Let\(X\) be a Banach space. A classical result due to Garkavi-Klee says that for every bounded set \(M\), the Chebyshex centre lies in the closed convex hull if and only if either X is a Hilbert space or of dimension atmost \(2\). Another classical result due to Amir-Lau is that for a uniformly convex space \(X\), a Hausdorff topological space \(Q\), evevry bounded set in the space of \(X\)-valued bounded continuous functions, \(C(Q,X)\) has a Chebyshec centre.
L. Vesely on the other hand proved that any non-reflexive Banach space can be equivalently renormed, to have a \(3\)-point set without a Chebyshev centre.
As for uniqueness, we recall Garkavi’s result that every compact set has atmost one Chebyshev centre if and only if the space is strictly convex. If this requirement holds for all bounded sets, then \(X\) is a URED space.
On properties of the Chebyshev centre map, we quote, Tsar’kov’ result, if \(\emptyset \neq V \subset \ell^{\infty}_n\) is a polyhedral subset, then it is a Lipschitz map on \(\ell^{\infty}_n\) . The behaviour of the Hausdorff metric on non-emppty bounded susbets of \(C(Q)\) was also studied by this author. A result of Pai and Nowroji considers this for closed subalgebras of \(C(Q)\)
It is known that in \(\ell^{\infty}(\Gamma)\), for a bounded set \(M\), the Chebyshev radius is equal to half the diameter and the Chebyshev centre map admits a \(1\)-Lipschitz selector. See the reviewers article [Proc. Am. Math. Soc. 130, No. 9, 2593–2598 (2002; Zbl 1007.41021)] for more on this.
I have bearely covered half of this long article, recommend, “full read” to all.
This article will become a ‘must read’ for any one interested in these aspects of approximation theory and its applications. The Russian school has originated and pioneered some of the fundamental concepts covered here. It continues to be a very active area. By bringing to the fore a lot of work (both classical and current) done by the European and Indian schools, the authors have done a great service.
For a bounded set \(M\), the Chebyshev centre, is the centre of a ball of smallest radius containing the set. Thus it is a point that best approximates the entire set.
Let \(Z(M)\) denote the set of Chebyshev centres of \(M\) and the set-valued map, \(M \rightarrow Z(M)\) is called the Chebyshev centre map. For a non-empty set \(Y \subset X\), \(Z_Y(M)\) denotes the set of relative Chebyshev centres. This is a convex set, when \(Y\) is.
Let\(X\) be a Banach space. A classical result due to Garkavi-Klee says that for every bounded set \(M\), the Chebyshex centre lies in the closed convex hull if and only if either X is a Hilbert space or of dimension atmost \(2\). Another classical result due to Amir-Lau is that for a uniformly convex space \(X\), a Hausdorff topological space \(Q\), evevry bounded set in the space of \(X\)-valued bounded continuous functions, \(C(Q,X)\) has a Chebyshec centre.
L. Vesely on the other hand proved that any non-reflexive Banach space can be equivalently renormed, to have a \(3\)-point set without a Chebyshev centre.
As for uniqueness, we recall Garkavi’s result that every compact set has atmost one Chebyshev centre if and only if the space is strictly convex. If this requirement holds for all bounded sets, then \(X\) is a URED space.
On properties of the Chebyshev centre map, we quote, Tsar’kov’ result, if \(\emptyset \neq V \subset \ell^{\infty}_n\) is a polyhedral subset, then it is a Lipschitz map on \(\ell^{\infty}_n\) . The behaviour of the Hausdorff metric on non-emppty bounded susbets of \(C(Q)\) was also studied by this author. A result of Pai and Nowroji considers this for closed subalgebras of \(C(Q)\)
It is known that in \(\ell^{\infty}(\Gamma)\), for a bounded set \(M\), the Chebyshev radius is equal to half the diameter and the Chebyshev centre map admits a \(1\)-Lipschitz selector. See the reviewers article [Proc. Am. Math. Soc. 130, No. 9, 2593–2598 (2002; Zbl 1007.41021)] for more on this.
I have bearely covered half of this long article, recommend, “full read” to all.
Reviewer: T.S.S.R.K. Rao (Bangalore)
MSC:
| 41A46 | Approximation by arbitrary nonlinear expressions; widths and entropy |
| 41A28 | Simultaneous approximation |
| 41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |
| 46B20 | Geometry and structure of normed linear spaces |
| 54C60 | Set-valued maps in general topology |
| 54C65 | Selections in general topology |
Keywords:
Chebyshev centre; Chebyshev-centre map; Chebyshev net; Chebyshev point; Jung constant; fixed point theorem; normal structure coefficientCitations:
Zbl 1007.41021Software:
MiniballReferences:
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