Let $(g\_k)$ be a sequence of independent standard gaussians variables on a fixed probability space $\Omega$. Let $(\epsilon\_k)$ be a sequence of independent rademacher variables.


What is the best constant in the following inequality:
$$
\vert\vert\sum\_{k}\epsilon\_k \otimes x\_k \vert\vert\_{L^2(\Omega,E)}
\leq 
K \vert\vert\sum\_{k}g\_k\otimes x\_k \vert\vert\_{L^2(\Omega,E)}\ \ ?
$$
for any Banach space $E$, any $x\_1,\ldots, x\_n \in E$.
I know that the best constant is $\leq \sqrt{\frac{\pi}{2}}$ (see Diestel,Jarchow,Tonge "Absolutely summing operators" page 239).



Best constant in comparison between Rademacher and gaussian averages?
$\sqrt{\pi\over 2}$, the reciprocal of the $L\_1$ norm of a standard gaussian, is the best constant. Let $x\_k$ be the kth unit basis vector in $\ell\_1$ and let the sum go from $1$ to $N$. The square of the left hand side is $N^2$ and the square of the right hand side is $N+N\sqrt{2\over \pi}(N-1)\sqrt{2\over \pi}$ (multiplied by $K^2$).


