Question

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.

Answer 1

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?

Answer 2

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 $n$.

Answer 3

Err ... yes. I meant to have normalized this, with each of the $\beta_i$'s divided by $\sqrt{n}$. Thanks for the quick responses!

Answer 4

A question somewhat less naive than the one I asked above is whether, given the constraints above on the $u_i$, there is a bound independent of $n$ on

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

where $\epsilon = ((\epsilon_1,\epsilon_2, ..., \epsilon_n))$ varies over all sequences of $\pm 1$. I suspect now that I've thought a bit more about it that this too is false, but it is at least a bit more reasonable guess.

I should perhaps have mentioned this earlier, but I was motivated to ask the question by some relevance this sort of thing has to a plank conjecture Keith Ball makes at the end of this paper. If what I've guessed here is false, then one could add in an upper-2 bound on the $u_i$'s of the sort Ball uses in the end of his paper.