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).