Theorem (Gaussian Source maximizes $R(D)$)

Suppose a continuous source $X$ has mean zero and variance $\sigma^{2}$. Considering the MSE distortion, the rate distortion function is upper bounded as $$R(D)\le\begin{cases} \frac{1}{2}\log \frac{\sigma^{2}}{D}&0<D\le \sigma^2 \\ 0&D>\sigma^{2} \end{cases}$$ i.e. $$R(D)\le R_{G}(D)$$