The cotype constant and an almost Euclidean decomposition for finite- dimensional normed spaces
In this paper it is shown that every 2n-dimensional normed space admits a ``Kashin-decomposition'', i.e. two n-dimensional subspaces \(E_ 1\) and \(E_ 2\), which are orthogonal with respect to the inner product induced by the John ellipsoid of E for which the Banach-Mazur distance to the Hilbert space \(\ell^ n_ 2\) is majorized by a function of the cotype 2 constant of E. This result has to be put in the context of previous results of Kashin (1977) on the decomposition of \(\ell_ 1^{2n}\) into two orthogonal Hilbertian spaces, and of Szarek (1978) which deduces the existence of Kashin decomposition from the boundedness of volume ratios; and of the more recent result of Milman and Pisier about the boundedness of the volume ratio under weak cotype 2 assumption. The authors show in fact that the set of [\(\theta\) n]-dimensional subspaces F of an n-dimensional space E, whose distances to Hilbert space are majorized by a certain function of cotype 2 constant \(K_ 2(E)\) of E, is of large measure (for the invariant probability on the Grassmannian manifold). This is a simple consequence of previous results of Szarek and of Tomczak-Jaegermann when one replaces \(K_ 2(E)\) by \(T_ 2(E^*)\) (type 2 constant of the dual) and one passes from \(T_ 2(E^*)\) to \(K_ 2(E)\) by a lemma inspired by the ``iteration procedure'' of V. D. Milman.
boundedness of the volume ratio under weak cotype 2 assumption
scientific article; zbMATH DE number 4073711
scientific article