Let $a$ be a vector in $R^m$, such that $\sum_{i=1}^ma_i=0$ I would like to bound $\sqrt{2m(2m-1)}\|a\|_{\infty}$ by $\sqrt{2m}\|a\|_2$ (or other way arround with the sharp constants), in the case when vector $a$ is compressible and when vector $a$ is incompressible:
(Here is definitions of the compressible and incompressible vectors I use: If we consider $R^{\sigma^c}=\mathsf{span}\{e_i: i\in\sigma^c\}$, $\sigma\subset\{1,...,m\}$, $|\sigma^c|=M$.
$x \in \mathsf{Comp}(M, \rho)$ iff there exists $\sigma$ with $|\sigma^c|\leq M$, s.t. projection $|P_{\sigma}x|\leq \rho$.
$x \in \mathsf{Incomp}(M, \rho)$ iff for any $\sigma$ with $|\sigma^c|\leq M$, s.t. projection $|P_{\sigma}x|> \rho$.
I wanted somehow to control $l_{\infty}$ norm. I guess I can do it only with incompressible vectors. But what can we say about compressible vestors?)
