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.



A bound on linear functionals over cotype 2 spaces
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?


