Hardy–Krause variation on subrectangles anchored at the lower corner

Let $f:[0,1{]}^d\to \mathbb{R}$ and let $R={\prod }_{j=1}^d[a_j,b_j]\subseteq [0,1{]}^d$ be a closed axis-parallel rectangle, allowing zero side lengths. For each nonempty subset $u\subseteq \{1,\dots ,d\}$, define the lower-face restriction $f_{u,0,R}(x_u):=f(z),z_j=\{\begin{array}{ll}{x}_j,& j\in u,\\ a_j,& j\mathrm{\notin }u\mathrm{.}\end{array}$ If every side indexed by $u$ is nondegenerate, its Vitali variation on ${\prod }_{j\in u}[a_j,b_j]$ is the supremum of the sums of absolute alternating increments over finite product grids on that rectangle, using the convention of Hardy–Krause variation anchored at the upper corner. If any side indexed by $u$ has length zero, that term is defined to be zero. The Hardy–Krause variation on $R$ anchored at its lower corner is $V_{HK,0}(f;R):={\sum }_{\mathrm{\varnothing }\mathrm{\ne }u\subseteq \{1,\dots ,d\}}{V}_V(f_{u,0,R};{\prod }_{j\in u}[a_j,b_j]),$ with value $+\mathrm{\infty }$ if any summand is infinite. Write $V_{HK,0}(f):=V_{HK,0}(f;[0,1{]}^d)$. The lower anchor fixes every complementary coordinate at the lower endpoint of $R$; this auxiliary convention is distinct from the upper anchor used in Hardy–Krause variation anchored at the upper corner.