An infinite-dimensional generalization of the Jung theorem
type: Item
instance of: scholarly article
title: An infinite-dimensional generalization of the Jung theorem
zbMATH Open document ID: 1123.46012
author: Nguyen Khac Viet
author: Nguyen Van Khiem
published in: Mathematical Notes
publication date: 2007-05-21T00:00:00Z
review text: For two nonempty subsets \(A,B\) of a Banach space \((X,\| \cdot\| ),\) with \(A\) bounded of diameter \(d(A)\), let \(r_B(A)=\inf_{y\in B}\sup_{x\in A}\| x-y\|\) denote the Chebyshev radius of the set \(A\) with respect to \(B\) and let \(r(A)\) be the Chebyshev radius with respect to cl\,co\((A)\), the closed convex hull of \(A\). The Jung constant \(J(X)\) and the relative Jung constant \(J_s(X)\) of the space \(X\) are defined by \(J(X)=\inf \{r_X(A): A \subset X\), \(d(A)=1\}\), respectively, \(J_s(X)=\inf \{r(A): A \subset X\), \(d(A)=1\}\). By a classical result proved by \textit{H. W. E. Jung} [J.~Reine Angew. Math. 123, 241--257 (1901; JFM 32.0296.05)] (see also \textit{R. Webster} [Convexity. Oxford: Oxford University Press (1994; Zbl 0835.52001)]), \(J(E^n)=J_s(E^n)= (\frac{n}{2(n+1)})^{1/2}\), for the \(n\)-dimensional Euclidean space \(E^n\). \textit{N. A. Routledge} [Q. J. Math. Oxf. (2) 3, 12--18 (1952; Zbl 0046.12301)] proved that for an infinite-dimensional Hilbert space \(H\), \(J(H)=J_s(H) =1/\sqrt 2\) [see also \textit{D. Amir}, Pac. J. Math. 118, 1--15 (1985; Zbl 0529.46011)]. A bounded subset \(A\) of \(X\) is called extremal (relatively extremal) if \(r_X(A)=J(X)\cdot d(A)\) (respectively, \(r_X(A)=J(X)\cdot d(A)\)). In the same paper, Jung proved that a bounded subset \(A\) of \(E^n\) is extremal if and only if it contains a regular \(n\)-simplex with edges of length \(d(A)\). In the case of a Hilbert space \(H\), one has that \(r(A)<(1/\sqrt{2})d(A)\), implying that the set \(A\) is not relatively compact. The aim of the present paper is to give a complete characterization of extremal subsets of a Hilbert space \(H\) in terms of the Hilbert and Kuratowski measures of noncompactness \(\chi(A)\) and \(\alpha(A)\), respectively. Let \(A\subset H\) with \(d(A)=\sqrt 2\). If \(A\) is extremal, then \(\chi(A)=1\) and, for every \(\varepsilon\in(0,\sqrt{2})\) and \(p\in \mathbb N\), there exists a \(p\)-simplex \(\Delta(p,\varepsilon)\) with vertic…
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46C05
Mathematics Subject Classification ID: 46C15
zbMATH DE Number: 5155519
zbMATH Keywords: Jung theorem
zbMATH Keywords: Jung constant
zbMATH Keywords: Chebyshev radius
zbMATH Keywords: Chebyshev center
zbMATH Keywords: measure of noncompactness
zbMATH Keywords: Hilbert spaces
reviewed by: Stefan Cobzaş
MaRDI profile type: Publication
full work available at URL: s11006-006-0131-6
OpenAlex ID: W2084193928
cites work: Normal structure coefficients for Banach spaces
cites work: Q5341481
cites work: A RESULT IN HILBERT SPACE
cites work: Q5570488
cites work: Q3684642
cites work: Q3814209
cites work: On Connections between Set and Ball Measures of Noncompactness
DOI: 10.1007/S11006-006-0131-6
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description: scientific article; zbMATH DE number 5155519