Almost Euclidean Subspaces of ℓ ⁿ_{p} Spaces, of General n-Dimensional Normed Spaces, and of Quotient of n-Dimensional Spaces

Abstract

In order to use Theorem 4.2. in special spaces one has to evalute M _r /b from below. It turns out that it is much easier to evaluate the average

$$ A_r = \int_{S^{n - 1} } {\parallel x\parallel d\mu (x)} $$

rather than the median M _r . The next lemma shows that typically the two quantities are close each to the other.