Question

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).

Answer 1

$\sqrt{\pi\over 2}$, 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 $K^2$).