Abstract
Let x1, x2, ..., xn be n unit vectors in a normed space X and define Mn=Ave{‖Σ(Formula presented.)ε1xi‖:ε1=±1}. We prove that there exists a set A⊂{1, ..., n} of cardinality(Formula presented.) such that {xi}i∈A is 16 Mn-isomorphic to the natural basis of l(Formula presented.). This result implies a significant improvement of the known results concerning embedding of l(Formula presented.) in finite dimensional Banach spaces. We also prove that for every ∈>0 there exists a constant C(∈) such that every normed space Xn of dimension n either contains a (1+∈)-isomorphic copy of l(Formula presented.) for some m satisfying ln ln m≧1/2 ln ln n or contains a (1+∈)-isomorphic copy of l(Formula presented.) for some k satisfying ln ln k>1/2 ln ln n−C(∈). These results follow from some combinatorial properties of vectors with ±1 entries.
Original language | English (US) |
---|---|
Pages (from-to) | 265-280 |
Number of pages | 16 |
Journal | Israel Journal of Mathematics |
Volume | 45 |
Issue number | 4 |
DOIs | |
State | Published - Dec 1983 |
Externally published | Yes |
All Science Journal Classification (ASJC) codes
- General Mathematics