On the modulus of noncompact convexity of a Banach space
Let \(A\) be a bounded subset of an infinite-dimensional Banach space \(X\). The Hausdorff measure of noncompactness \(\chi_ A\) of the set \(A\) is the infimum of all numbers \(r> 0\) such that \(A\) can be covered by finitely many balls of radius \(r\). For \(\varepsilon\in [0, 1]\), let \[ \Delta(\varepsilon)= \inf\{1- \inf\{\| x\|: x{\i}A\}\}, \] where the infimum is taken over all nonempty closed convex subsets \(A\) of the unit ball \(B_ X\) of \(X\) with \(\chi_ A\geq \varepsilon\). The Kuratowski measure of noncompactness of a bounded subset \(A\) of \(X\) is defined as the infimum of all numbers \(d> 0\) such that \(A\) can be covered by finitely many sets with diameters not exceeding \(d\). Let \(\Delta_ K\) be a modulus whose definition is obtained from the definition of \(\Delta\) by replacing the measure \(\chi\) by the Kuratowski measure. The function \(\Delta_ K\) is defined on the interval \([0, 2]\), and \(\Delta_ K(\varepsilon)\leq \Delta(\varepsilon)\leq \Delta_ K(2\varepsilon)\) for every \(\varepsilon\in [0, 1]\). Some properties of the modulus \(\Delta\) are known in the literature. In this paper the author establishes counterparts of some basic properties of the modulus of convexity for the moduli \(\Delta\) and \(\Delta_ K\) and proves a result which shows a difference between these notions viz. a Hilbert space has the best (i.e. the largest) possible modulus of convexity. He also shows that there is no space with the best modulus of non-compact convexity which gives answers to questions raised by \textit{J. Banas} [Nonlinear Anal., Theory Methods Appl. 16, No. 7/8, 669-682 (1991; Zbl 0724.46019)] and \textit{T. Sekowski} [Rend. Sem. Mat. Fis. Milano 56, 147-153 (1986; Zbl 0655.47050)].
scientific article; zbMATH DE number 701022
scientific article