Given a set of vectors, $A\subset R^n$, we define
$R(A) = \frac{1}{N} \mathbb{E}_\sigma \left[ \sup_{a\in A} \sum_{i=1}^N\sigma_i a_i\right]$,
where $\sigma$ is Rademacher random variable such that $P(\sigma=1) =P(\sigma =-1) =\frac{1}{2}$. Also, define
$\tilde{R}(A) = \frac{1}{N} \mathbb{E}_\sigma \left[ \sup_{a\in A} \beta\sigma_1a_1+\sum_{i=2}^N\sigma_i a_i\right]$,
where $\beta \ge 1$. If we further assume that for any $a\in A$, we also have $-a \in A$. Then, I was wondering, if we can prove
$\tilde{R}(A) \ge R(A)$?
Thanks!