Comparing norms of a vector

Let $a$ be a vector in $\mathbb R^m$, such that $\sum_{i=1}^{m}a_i=0.$

I would like to compare $\sqrt{2m(2m−1)}\|a\|_{\infty}$ and $\sqrt{2m}\|a\|_2$, in the case when the vector $a$ satisfies the following:

Consider $R^{σ^c}=\operatorname{span}\{e_i:i\in\sigma^c\}, \sigma\subset \{1,\ldots,m\}, |\sigma^c|=M$. ($\sigma^c$ is a complement of $\sigma$). There exists $σ$ with $|σ^c|≤M$, such that projection $|P_σa|≤ρ$, for $\rho>0$.

Of course, there is an inequality $\|a\|_2\leq \sqrt m\|a\|_{\infty}$. But vector $a$ which satisfy condition above has few big (in absolute value) coordinates and the most of the coordinates are close to zero. I would like to get something better.

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Thank you. Its a typo – Nick G.H. Mar 30 '12 at 16:46
First you say that $|\sigma^c| = M$, then you say that $|\sigma^c|\leq M$. Which one is it? Anyway, the best you can do is $\|a\|_2\leq \sqrt{m - M}\|a\|_\infty$. Why is this not obvious? Am I missing something? – William Mar 30 '12 at 18:30