On the spread of positively curved Alexandrov spaces (Q2249629)

From the author's abstract: ``It was proved by \textit{F. H. Wilhelm, jun.} [Invent. Math. 107, No. 3, 653--668 (1992; Zbl 0729.53044)] that Gromov's filling radius of closed positively curved manifolds [\textit{M. Gromov}, J. Differ. Geom. 18, 1--147 (1983; Zbl 0515.53037)] with a uniform lower bound on sectional curvature attains its maximum with the round sphere. Recently the author proved [Math. Z. 273, No. 1--2, 161--171 (2013; Zbl 1266.53045)] that this is also the case for closed finite-dimensional Alexandrov spaces with a positive lower curvature bound. This was proved as a corollary of a comparison theorem for the invariant called spread of those spaces. In this paper, we extend the latter result to infinite-dimensional Alexandrov spaces.'' From the introduction: ``In the present paper, which is a continuation of the previous paper [the author, op. cit.], we are concerned with the geometry of possibly infinite-dimensional Alexandov spaces with a positive lower curvature bound. As the nature of infinite-dimensional Alexandrov spaces has not been fully explored, we are particularly interested in those spaces. Our goal is to establish a comparison theorem for them with the round sphere as the model space, which extends the main result of [ibid.].'' For a metric space \(X=(X,d)\), its spread is defined [\textit{F. Wilhelm}, op. cit.], denoted by \(\mathrm{Spread}(X)\), as the infimum of all \(\lambda>0\) for which there is a subset \(Y\subset X\) of diameter \(\mathrm{Diam}(Y)\leq\lambda\), such that \(d(x,Y)\leq\lambda\) for any \(x\in X\). A metric space is an ``Alexandrov space of curvature \(\geq k\)'' for some real number \(k\in R\), or simply an ``Alexandov space'', if it is a complete length space, not necessarily a geodesic space, whose distance function enjoys the so-called ``quadruple condition'' [the author, op. cit]. The main result of the paper is the following theorem. {Theorem 5}. Any infinite-dimensional Alexandrov space \(X\) of curvature \(\geq 1\) fulfills \(Spread(X)\leq \pi/2\). This theorem extends a theorem of the author for the case of finite-dimensional Alexandrov spaces of curvature \(\geq 1\) [the author, op. cit.], which in turn extends a theorem of F. Wilhelm stating that for any \(n\)-dimensional closed Riemannian manifold \(V\) of sectional curvature \(\geq 1\), either \(\mathrm{Spread}(V)<\mathrm{Spread}(\mathbb{S}^n)\) or \(V\) is isometric to the round sphere \(\mathbb{S}^n\).