$\sqrt{\pi\over 2}$, the reciprocal of the $L_1$ norm of a standard gaussian, is the best constant. Let $x_k$ be the kth unit vector 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 $K^2$).
$\sqrt{\pi\over 2}$, the reciprocal of the $L_1$ norm of a standard gaussian, is the best constant. Let $x_k$ be the kth unit vector basis 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 $K^2$).