Edge-isoperimetric problem for the three-dimensional and higher-dimensional [?]ell-infinity lattices

Solve the edge-isoperimetric problem for the Cayley graphs [?]((\mathbb{Z}^d,\ell_\infty)[?] in dimensions \(d\geq 3\), determining the minimum edge boundary of every finite vertex subset of prescribed cardinality.

Background

The paper discusses the edge-isoperimetric problem (EIP), which asks for the minimum number of edges leaving a vertex subset of fixed size. For the graph $(\mathbb{Z}^2,\ell_\infty)$, Brass solved the EIP and showed that nested optimal solutions exist.

The corresponding problem for $(\mathbb{Z}^d,\ell_\infty)$ in dimensions $d\geq 3$ is identified as unresolved. The open problem concerns determining the optimal shapes or edge-boundary values for finite subsets in these higher-dimensional lattice graphs.