Counterexample Cayley graph on Z^d (d≥2) without nested edge-isoperimetric minimizers, plus positive example on a triangular lattice variant.
Vesztergombi,Bounds on the number of small distances in a finite planar set, Studia Sci
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Edge isoperimetry of lattices
Counterexample Cayley graph on Z^d (d≥2) without nested edge-isoperimetric minimizers, plus positive example on a triangular lattice variant.