Let $\mathcal{Q}_{N}$ denote the family of all $N$-level quantizers. $Q^{*}\in\mathcal{Q}_{N}$ is an optimal quantizer if $$E[d(X,Q^{*}(X))]=\min_{Q\in\mathcal{Q}_{N}}E[d(X,Q(X))]$$ ## Note $Q^{*}$ is not necessarily unique. $Q^{*}$ exists for all $N$ and “reasonable” $d$ if $E[d(X,y)]<\infty$ for some $y\in\mathbb{R}$.