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Given a $k$-dimensional subspace in $\ell^n_p$, is there a way to bound the value of $$ \sum_{i=1}^k \|a_i\|_{\ell^p}^2 $$ for $a_i$ an orthonormal (for the "standard" underlying $\ell^n_2$) basis.

More precisely, are there estimates for the average (for the usual probability measure on $O(k)$) value or the minimal one.

EDIT: Some results using martingales or volume ratios seem to produce estimates for similar quantities, which made me believe there are known estimates for this.

Also, it seems the case where this methods apply is when $p \leq 2$, so this should be an added to the hypothesis.

EDIT: Of course, when $k=1$, one may use Hölder's inequality to get that it is $\leq n^{\frac{2}{p}-1}$ (bound attained when $X$ is spanned by a vector with all coefficients equal).

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up vote 4 down vote accepted

I think Lewis' lemma is relevant. It says that if $E$ is a $k$ dimensional subspace of $L_p(\mu)$ then there is a change of density $g$ s.t. $M_{g,p}E$ (which is normalized to make $M_{g,p}$ an isometry from $L_p(\mu)$ onto $L_p(g d\mu)$) has a basis $f_1, \dots, f_k$ that is orthonormal in $L_2(g d\mu)$ and $\sum_{j=1}^k |f_j|^2 =k$. The relevant D. R. Lewis papers are in Studia 63 (1978) and Mathematica 26 (1979). Or look at section 1.2 of my article with Schechtman in the Handbook of the Geometry of Banach Spaces. If you are not willing to change to an equivalent measure, then the worst case occurs when the subspace is spanned by $k$ elements of the unit vector basis. BTW: It is more natural and simpler when comparing norms to work with the uniform probability measure on $\{1,\dots,n\}$ rather than counting measure. At the end you can always translate back to counting measure.

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One application of Lewis' lemma is that if $E$ is a $k$-dimensional subspace of $L_p(\mu)$, then the Banach-Mazur distance ($=$ Euclidean distortion $=$ isomorphism constant) to $\ell_2^k$ is at most $k^{|1/p -1/2|}$, a bound that is achieved for $E = \ell_p^k$. –  Bill Johnson Sep 24 '13 at 15:31
    
Thanks for these nice references (though I must confess not being able to conclude). There is one thing that puzzles me: in the question, the worst case seems rather to be when $X$ is spanned by $k$ disjointly supported blocks made of $m$ $1$s (with $km=n$, this gives a $k \cdot m^{2/p-1}$) and not when $X$ is spanned by unit vectors (which gives $k$). –  Tonio Sep 24 '13 at 18:59
    
By worst I meant the condition number (isomorphism constant) of the formal identity $I_E: (E,\|\cdot\|_p) \to (E,\|\cdot\|_2)$. –  Bill Johnson Sep 24 '13 at 20:04

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