$Q$ is a nearest neighbour vector quantizer if for all $\mathbf{x}$ $$Q(\mathbf{x})=\underset{ \mathbf{c}_{j}\in\mathcal{C}}{ \arg\min }\ d(\mathbf{x},\mathbf{c}_{j})$$or equivalently for all $i=1,\dots,N$ $$R_{i}\subset \{ \mathbf{x}:d(\mathbf{x},\mathbf{c}_{i})\le d(\mathbf{x},\mathbf{c}_{j}),j=1,\dots,N \}$$