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

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 [Kal1, Kal2, 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 compactum [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 qand quasi-normed spaces. Recall that a quasi-norm on a real vector space X is a map ‖ · ‖ : X → R satisfying these conditions: (1) ‖x‖ > 0 for all x 6= 0. (2) ‖tx‖ = |t|‖x‖ for all t ∈ R and x ∈ X. (3) There exists C ≥ 1 such that ‖x + y‖ ≤ C(‖x‖+ ‖y‖) for all x, y ∈ X. If (3) is substituted by (3a) ‖x + y‖q ≤ ‖x‖q + ‖y‖q for all x, y ∈ X, for some fixed q ∈ (0, 1], This research was supported by Grant No. 92–00285 from United States–Israel Binational Science Foundation (BSF).