Lower bound for $\|x-y\|$

Question

For $x$,$y$ in Hilbert space $\mathcal{H}$ I want a lower bound for \begin{equation} \|x-y\|_{\mathcal{H}}^2 \end{equation} I know \begin{equation} |\ \|x\|_{\mathcal{H}}-\|y\|_{\mathcal{H}}\ |\leq\|x-y\|_{\mathcal{H}} \end{equation} However is there a better answer than this? My Hilbert space is $L^2$. Any help is greatly appreciated.