Embedding ofl∞k in finite dimensional Banach spaces

AbstractLetx1,x2, ...,xn ben unit vectors in a normed spaceX and defineMn=Ave{‖Σi=1nε1xi‖: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 {xi}i∈A is 16Mn-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 spaceXn of dimensionn either contains a (1+∈)-isomorphic copy ofl2m 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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