The Extension of the Finite-Dimensional Version of Krivine's Theorem to Quasi-Normed Spaces

Summary

In 1980 D. Amir and V. D. Milman gave a quantitative finitedimensional version of Krivine's theorem. We extend their version of the Krivine's theorem to the quasi-convex setting and provide a quantitative version for p-convex norms.

Recently, a number of results of the Local Theory have been extended to the quasi-normed spaces. There are several works [KalI, Ka12, D, GL, KT, GK, BBP1, BBP2, M2] where such results as Dvoretzky-Rogers lemma [DvR], Dvoretzky theorem [Dv1, Dv2], Milman's subspace-quotient theorem [M1], Krivine's theorem [Kr], Pisier's abstract version of Grotendick's theorem [P1, P2], Gluskin's theorem on Minkowski compact urn [G], Milman's reverse Brunn-Minkowski inequality [M3], and Milman's isomorphic regularization theorem [M4] are extended to quasi-normed spaces after they were established for normed spaces. It is somewhat surprising since the first proofs of these facts substantially used convexity and duality.

In [AM2] D. Amir and V. D. Milman proved the local version of Krivine's theorem (see also [Gow], [MS]). They studied quantitative estimates appearing in this theorem. We extend their result to the q- and quasi-normed spaces. Recall that a quasi-norm on a real vector space X is a map ||·|| : X → ℝ⁺ satisfying these conditions: Note that I-norm is the usual norm. It is obvious that every q-norm is a quasi-norm with C = 2 ^{1 / q-1} . However, not every quasi-norm is q-norm for some q. Moreover, it is even not necessary continuous. It can be shown by the following simple example.