\begin{proof} $$\begin{align*} \frac{1}{k}D(Q)&=\frac{1}{k}\sum_{i=1}^{N}\int\limits _{R_{i}}\lVert \mathbf{x}-\mathbf{c}_{i} \rVert ^{2}f(\mathbf{x}) \, d\mathbf{x}&(1)\\ &\approx \frac{1}{N^{2/k}}\sum_{i=1}^{N}G(R_{i}) \frac{f(\mathbf{c}_{i})}{\lambda(\mathbf{c}_{i})^{2/k}}V(R_{i})&(2)\\ &\approx \frac{G(R_{k}^{*})}{N^{2/k}}\sum_{i=1}^{N} \frac{f(\mathbf{c}_{i})}{\lambda(\mathbf{c}_{i})^{2/k}}V(R_{i})&(3)\\ &\approx \frac{C_{k}^{*}}{N^{2/k}}\int\limits _{\mathbb{R}^{k}}\frac{f(\mathbf{x})}{\lambda(\mathbf{x})^{2/k}} \, d\mathbf{x}&(4)\\ &\approx N^{-2/k}C_{k}^{*}\lVert f \rVert _{\frac{k}{2+k}}&(5) \end{align*}$$With (1) being by definition. With (2) we have that $\lambda$ is the point density of the sequence of quantizers that satisfies: For any reasonable $R\subset \mathbb{R}^{k}$, $$\lim_{ N \to \infty } \frac{1}{N}(\text{\# of codevectors in }R)=\int\limits _{R}\lambda(\mathbf{x}) \, d\mathbf{x}$$where $\lambda$ is a pdf. and: $$\int\limits _{R_{i}} \, d\mathbf{x}\approx G(R_{i})V(R_{i})$$ With (3) and (4) we apply
With (5) we apply lemma 4. Where optimal $\lambda$ satisfies: $$\lambda(\mathbf{x})=\frac{f(\mathbf{x})^{k/(2+k)}}{\lVert f \rVert _{{k}/{(2+k)}}^{k/(2+k)}}$$ \end{proof}