An infinite-dimensional generalization of the Jung theorem
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 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 \(\alpha (A) =\sqrt 2\).
scientific article; zbMATH DE number 5155519
scientific article