Three-dimensional box norm ($\Vert f{\Vert }_{{\square }^3}$)

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.