Abstract
Letx 1,x 2, ...,x n ben unit vectors in a normed spaceX and defineM n =Ave{‖Σ n i=1 ε1 x i ‖:ε1=±1}. We prove that there exists a setA⊂{1, ...,n} of cardinality\(\left| A \right| \geqq \left[ {\sqrt n /\left( {2^7 M_n } \right)} \right]\) such that {x i } i∈A is 16M n -isomorphic to the natural basis ofl k∞ . This result implies a significant improvement of the known results concerning embedding ofl k∞ in finite dimensional Banach spaces. We also prove that for every ∈>0 there exists a constantC(∈) such that every normed spaceX n of dimensionn either contains a (1+∈)-isomorphic copy ofl m2 for somem satisfying ln lnm≧1/2 ln lnn or contains a (1+∈)-isomorphic copy ofl k∞ for somek satisfying ln lnk>1/2 ln lnn−C(∈). These results follow from some combinatorial properties of vectors with ±1 entries.
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The contribution of the first author to this paper forms part of his Ph.D. Thesis written under the supervision of Prof. M. A. Perles from the Hebrew University.
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Alon, N., Milman, V.D. Embedding ofl k∞ in finite dimensional Banach spaces. Israel J. Math. 45, 265–280 (1983). https://doi.org/10.1007/BF02804012
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DOI: https://doi.org/10.1007/BF02804012