Hardy Krause variation n-dimensions

We follow Chapter 2 of Niederreiter [#!kn:Niederreiter1992!#] page 19. We consider the s dimensional unit hypercube. Let J be a subinterval of the s dimensional unit hypercube. Let $\Delta(h;J)$ be the sum of values of h at the vertices of J with alternating signs at the vertices.

Definition 9 (Vitali Variation)   The variation of h on the unit hyper cube in the sense of Vitali is defined by

$$V^s(h) = sup_{P} \sum_{J in P} \vert\Delta(h;J)\vert$$ (2.5)

where the supremum is over all partitions P of the unit hypercube into subintervals.

Definition 10 (Hardy-Krause Variation)   The variation of h on the unit hyper cube in dimension s is

$$V(h) = \sum V^s(h)$$ (2.6)

where the sum is over the "faces" of the unit hypercube in s or lower dimensions.