Summary
Consider a sequence(x i ) i≦n of norm one vectors in a Banach space. For a subsetJ of {1,...,n} consider the equivalence constant ξ(J) between(x i ) i∈J and the ℓ1 basis, and consider ξ(k)=min{ξ(J); cardJ=k}. We give a near optimal relationship between the rate of decay of ξ(k) and the averageE of\(\left\| {\sum\nolimits_{i \leqq n} {\varepsilon _i x_i } } \right\|\) over all choices of signs. In particular, we show that one can choosek such that, for some universal constantK, k≧E 2/Kn and\(\xi (k) \geqq \left( {\frac{{Kn}}{{E^2 }}k} \right)^{ - 1/2} \left( {\log \left( {\frac{{Kn}}{{E^2 }}k} \right)} \right)^{ - K} \). This is optimal within the logarithmic term. We also prove, that forp<2, the notions of type and infratype coincide.
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Oblatum 27-III-1991
Work partially supported by an NSF grant
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Talagrand, M. Type, infratype and the Elton-Pajor theorem. Invent Math 107, 41–59 (1992). https://doi.org/10.1007/BF01231880
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DOI: https://doi.org/10.1007/BF01231880