Three-dimensional box norm
= Three-dimensional box norm
{title2=$\|f\|_{\square^3}$}
For a complex function on $X_1\times X_2\times X_3$, its eighth power is the average of the conjugated product over the eight vertices of a coordinate cube. Three successive applications of the <Cauchy-Schwarz inequality> show that it bounds correlation with a product of three bounded functions, each omitting one coordinate. This is the analytic estimate behind the <tetrahedron counting lemma>.