The normalized second moment of a bounded measurable set 
of positive volume 
is the scale-invariant quantity
 |
where 
is the geometric centroid of 
and 
is the Euclidean norm
(Conway and Sloane 1982). It is unchanged by translation,
orthogonal transformation, and positive
rescaling. The factor 
measures squared error per coordinate, while 
removes the dependence on scale.
For a point lattice 
spanning 
, its normalized second moment 
is that of its Voronoi cell
at the origin. This cell has geometric
centroid 0 and volume equal to the covolume
of 
.
The quantity 
measures the mean squared error of a lattice quantizer
at fixed covolume.
