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This is a modification of the somewhat naive question that I asked below.

Suppose $X$ is a real Banach space of cotype-2, and $u_1, u_2, ... u_n$ are unit vectors in this space. For $\gamma = ((\gamma_1, \gamma_2, ..., \gamma_n))$ an arbitrary element of $\mathbb{R}^n$, suppose that $\|\gamma_1 u_1 + \gamma_2 u_2 + \cdots \gamma_n u_n\| \geq \|\gamma\|_{\ell^\infty}$. Then is there a bound independent of $n$ (but depending on the cotype constant) of

$$\inf_\epsilon \sup_{\beta \geq 0} \frac{\sum \beta_i\cdot \frac{1}{\sqrt{n}}}{\|\sum \beta_i \epsilon_i u_i \|}$$

where $\epsilon = ((\epsilon_1,\epsilon_2, ..., \epsilon_n))$ varies over all sequences of $\pm 1$, and $\beta \geq 0$ means that $\beta_i \geq 0$ for all $i$.

My suspicion is that this is actually false, but I haven't constructed an explicit counterexample.

The original question I asked was resolved by two answers below. (I had left out a $\sqrt{n}$ term that I meant to include, but even with this, the answer is no.)

Suppose $X$ is a real Banach space of cotype-2, and $u_1, u_2, ... u_n$ are unit vectors in this space. For $\gamma = ((\gamma_1, \gamma_2, ..., \gamma_n))$ an arbitrary element of $\mathbb{R}^n$, suppose that $\|\gamma_1 u_1 + \gamma_2 u_2 + \cdots \gamma_n u_n\| \geq \|\gamma\|_{\ell^\infty}$. (It is important that this inequality be strict, not just an 'up to a constant' sort of bound. What is going on becomes more clear if a picture is drawn.) Is there a bound independent of $n$ on

$\sup_\beta \frac{\sum \beta_i}{\|\beta_1 u_1 + \cdots + \beta_n u_n\|}?$

If not, what if we impose the additional condition that the $\beta_i$ all be non-negative? One might guess the bound is linear with the cotype-2 constant.

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@Brad: I made an addition to my answer that you might find useful. –  Bill Johnson May 10 '10 at 9:10

2 Answers 2

up vote 3 down vote accepted

Did you mean to have a factor of ${\sqrt n}$ on the right hand side? Consider $u_i$ orthonormal in a Hilbert space.

In fact, the best constant is of order $n$, but if you restrict the $u_i$ to be C-unconditional, then obviously the sup is bounded by C times the cotype 2 constant times ${\sqrt n}$.

Further comments:

(1) Your inequality says that the $u_i$ are an Auerbach basis for the space they span, which means that there are functionals $u_i^*$ biorthogonal to $u_i$ and having norm one. If you have the inequality but only up to the constant $1/C$, the biorthogonal functionals have norm at most $C$. You can renorm the space with $(1/C\|x\|) \vee \max_i|u_i^*(x)|$ to make the $u_i$ an Auerbach basis--at worst this multiplies the cotype 2 constant by $C$.

(2) You can replace the top of your left hand side with $\sum |\beta_i|$ because your hypotheses on the $u_i$ does not change if you multiply some of them by $-1$. To see that you cannot do better than order $n$ as an upper estimate, consider $u_i :=(1/2)(e_i-e_{i+1})$ in $\ell_1$.

Added May 2, 2010:

In your modified question, you get a bound independent of $n$ under the stronger hypothesis that $X^*$ has type 2. Indeed, let $u_k^*$ be the (norm one) biorthogonal functionals to $u_k$ and choose signs $\epsilon_k$ so that $y^*:= n^{-1/2}\sum_{k=1}^n \epsilon_k u_k^*$ has norm at most $T_2(X^*)$. Then for any $\beta_k\ge 0$, $\|\sum_{k=1}^n \epsilon_k\beta_k u_k\|$ is at least as large as $T_2(X^*)^{-1} n^{-1/2} \sum_{k=1}^n \beta_k$.

Since for an $n$ dimensional space $E$, $T_2(E^*)\le C \log n C_2(E)$, you do get at worst a $\log n$ growth rate of your ratio.

In your revised question, do you know it is important that the $u_k$ form an Auerbach basis for their span?

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Thanks for this very helpful response. I tried to post a follow up question in this space, but was held up by character limits, so I've posted it below. –  Brad Rodgers Apr 26 '10 at 18:34
    
Aha! This is great to find out! (Sorry I didn't respond sooner, but I wasn't aware of the update.) In answer to your last question, what I ultimately want to prove is that the above inf-sup quantity is bounded when we have some constant $A$ independent of $n$ so that $\|\gamma_1 u_1 + \gamma_2 u_2 + \cdots \gamma_n u_n\| \leq A\|\gamma\|_{\ell^2}$. This may seem to go in the opposite direction as the criterion I give above, but both impose the constraint that boundary of the unit ball of our Banach space must be somewhat close to meeting each $u_i$ orthogonally. (For large $n$, at any rate.) –  Brad Rodgers May 13 '10 at 19:50
    
Sorry; I had to learn on meta how to make a changed answer move up the thread. So the condition you have is that the formal identity from $\ell_2^n$ into $\ell_\infty^n$ factors through your space. This condition was studied by Pelczynski and somebody (maybe Schutt?). I am traveling and cannot check references or use MathSciNet. –  Bill Johnson May 14 '10 at 10:54

No, not even with positive coefficients. Take, for instance, the $u_i$ to be orthonormal vectors in Hilbert space (which has type 2). Then the condition $\| \gamma \|_{\infty} \le \|\gamma_1 u_1 + \gamma_2 u_2 + \cdots + \gamma_n u_n\|$ clearly holds, whereas taking all $\beta_i$ equal to one gives a quotient $\frac{n}{\sqrt{n}} = \sqrt{n}$, definitely not bounded independently of $n$.

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Sorry to have crossed paths, Alejandro. What are you doing up so early? :) –  Bill Johnson Apr 26 '10 at 14:54

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