Type, infratype and the Elton-Pajor theorem

Abstract

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.