Coordinate compression in a product of paths

Coordinate compression in direction $i$ replaces every $i$-section of a subset of $[k{]}^d$ by the initial segment of the simplicial order on a grid having the same size. Assuming the vertex-isoperimetric theorem one dimension lower, this operation preserves cardinality and does not increase the external vertex boundary.

Let $A_j\subseteq [k{]}^{d-1}$ be the section of $A\subseteq [k{]}^d$ with one coordinate fixed at $j$, and put $A_0=A_{k+1}=\varnothing$. The corresponding section of its closed graph neighbourhood isAfter coordinate compression, the three sets on the right are nested initial simplicial segments. Their union therefore has the largest of their three cardinalities, which is no larger than the original union.