Nguyen Khac Viet; Nguyen Van Khiem An infinite-dimensional generalization of the Jung theorem. (English) Zbl 1123.46012 Math. Notes 80, No. 2, 224-232 (2006); translation from Mat. Zametki 80, No. 2, 231-239 (2006). 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 clco\((A)\), 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 H. W. E. Jung [J. Reine Angew. Math. 123, 241–257 (1901; JFM 32.0296.05)] (see also 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 \(E^n\). 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 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 vertices in \(A\) and edges of length \(\sqrt 2-\varepsilon\). Conversely, the existence of such simplexes implies that \(A\) is extremal. Moreover, if \(A\subset H\), \(r(A) =1\), is extremal, then \(\alpha (A) =\sqrt 2\). Reviewer: Stefan Cobzaş (Cluj-Napoca) Cited in 2 Documents MSC: 46B20 Geometry and structure of normed linear spaces 46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) 46C15 Characterizations of Hilbert spaces Keywords:Jung theorem; Jung constant; Chebyshev radius; Chebyshev center; measure of noncompactness; Hilbert spaces Citations:Zbl 0835.52001; Zbl 0046.12301; Zbl 0529.46011; JFM 32.0296.05 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] W. L. Bynum, ”Normal structure coe.cients for Banach spaces,” Pacific J. Math., 86 (1980), 427–436. · Zbl 0442.46018 [2] A. L. Garkavi, ”On Chebyshev centers and convex hulls of set,” Uspekhi Mat. Nauk [Russian Math. Surveys], 19 (1964), no. 6, 139–145. [3] H. W. E. Jung, ”Über die kleinste Kugel, die eine räumliche Figur einschliesst, ” J. Reine Angew. Math., 123 (1901), 241–257. · JFM 32.0296.05 · doi:10.1515/crll.1901.123.241 [4] L. Danzer, B. Grunbaum, and V. Klee, Helly’s Theorem and Its Relatives, Amer. Math. Soc., Providence, R. I., 1963; Russian transl.: Mir, Moscow, 1968. · Zbl 0132.17401 [5] N. A. Routledge, ”A result in Hilbert space,” Quart. J. Math., 3 (1952), no. 9, 12–18. · Zbl 0046.12301 · doi:10.1093/qmath/3.1.12 [6] V. I. Berdyshev, ”A relationship between the Jackson inequality and a geometric problem,” Mat. Zametki [Math. Notes], 3 (1968), no. 3, 327–338. · Zbl 0179.45602 [7] J. Daneš, ”On the radius of a set in a Hilbert space,” Comment. Math. Univ. Carolin., 25 (1984), no. 2, 355–362. · Zbl 0568.46018 [8] N. M. Gulevich, ”The radius of a compact set in a Hilbert space,” Zap. Nauchn. Sem. LOMI [J. Soviet Math.], 164 (1988), 157–158. · Zbl 0664.46023 [9] J. R. L. Webb and W. Zhao, ”On connections between set and ball measures of non-compactness,” Bull. London Math. Soc., 22 (1990), 471–477. · Zbl 0732.47049 · doi:10.1112/blms/22.5.471 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.