Definition ($G(R)$: Dimensionless normalized second moment of $R$)

Assume $R$ is a $k$-dimensional polytope with finite volume $V(R)$ with centroid at origin $$\min_{\mathbf{y}}\int\limits _{R}\lVert \mathbf{x}-\mathbf{y} \rVert ^{2} \, d\mathbf{x}=\int\limits _{R}\lVert \mathbf{x} \rVert ^{2} \, d\mathbf{x}$$The dimensionless normalized second moment of $R$ (a basic cell) is $$G(R)\triangleq \frac{1}{k} \frac{1}{V(R)^{1+2/k}}\int\limits _{R}\lVert \mathbf{x} \rVert ^{2} \, d\mathbf{x}$$

Definition (High-resolution conditions for LVQ)

Assuming High-Resolution Conditions (i.e. $\mathbf{X}\sim f$ where $f$ smooth and basic cell $R_{0}$ small enough) we have that the LVQ Distortion can be approximated as $$D(Q_{\Lambda})\approx \frac{1}{V(R_{0})}\int\limits _{R_{0}}\lVert \mathbf{x} \rVert ^{2} \, d\mathbf{x}$$and using the dimensionless second moment of $R_{0}$ we can redefine it as $$D(Q_{\Lambda})\approx V(R_{0})^{2/k}G(R_{0})$$

Definition (Minimum normalized second moment)

We define the minimum normalized second moment for $k$-dimensional lattices as $$G_{k}=\min_{\Lambda \subset \mathbb{R}^{k}}G(R_{0})$$