Abstract
We prove that every n-dimensional normed space with a type p < 2, cotype 2, and (asymptotically) extremal Euclidean distance has a quotient of a subspace, which is well isomorphic to \(\ell_p^k\) and with the dimension k almost proportional to n. A structural result of a similar nature is also proved for a sequence of vectors with extremal Rademacher average inside a space of type p. The proofs are based on new results on restricted invertibility of operators from \(\ell_r^n\) into a normed space X with either type r or cotype r.
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© 2003 Springer-Verlag Berlin/Heidelberg
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Schechtman, G., Tomczak-Jaegermann, N., Vershynin, R. (2003). Maximal \(\ell_p^n\)-Structures in Spaces with Extremal Parameters. In: Milman, V.D., Schechtman, G. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 1807. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-36428-3_18
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DOI: https://doi.org/10.1007/978-3-540-36428-3_18
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