Let $\mathcal{Q}_{N}$ denote the family of all $k$-dimensional $N$-point vector quantizers. $Q^{*}\in\mathcal{Q}_{N}$ is an optimal vector quantizer for source $\mathbf{X}$ if $$\mathbb{E}[d(\mathbf{X},Q^{*}(\mathbf{X}))]=\min_{Q\in\mathcal{Q}_{N}}\mathbb{E}[d(\mathbf{X},Q(\mathbf{X}))]$$