Full Length ArticleExact asymptotic volume and volume ratio of Schatten unit balls
Section snippets
Proof of Theorem 1
We start with the following observation. The equality was established by Saint Raymond [25, Corollaire 4 on p. 69]. For , which is always assumed in the following, Saint Raymond [25, p. 70] characterized the constant as the limit of , as , where He then showed that the positive sequence is decreasing, which implies that it converges to a limit, as , denoted by . By providing bounds on this
Proof of Theorem 2
Let us recall some definitions and provide some additional preliminaries. Let be a real -dimensional Banach space with unit ball . If we are given a complex Banach space, we ignore the complex structure and consider the space as a real one, so that is the dimension over . We denote by the (unique) maximal volume ellipsoid that is contained in . The volume ratio of is then defined as where stands for the usual -dimensional Lebesgue measure. Note
Acknowledgments
ZK has been supported by the German Research Foundation under Germany’s Excellence Strategy EXC 2044 – 390685587, Mathematics Münster: Dynamics - Geometry - Structure. JP has been supported by a Visiting International Professor Fellowship from the Ruhr University Bochum and its Research School PLUS and by the Austrian Science Fund (FWF) Project P32405 “Asymptotic Geometric Analysis and Applications”. ZK and CT have been supported by the DFG Scientific Network Cumulants, Concentration and
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