Edge-isoperimetric inequalities in the grid

The grid graph is the graph on [k]ⁿ={0,...,k–1}ⁿ in whichx=(x_i)₁ⁿ is joined toy=(y_i)₁ⁿ if for somei we have |x_i–y_i|=1 andx_j=y_j for allji. In this paper we give a lower bound for the number of edges between a subset of [k]ⁿ of given cardinality and its complement. The bound we obtain is essentially best possible. In particular, we show that ifA[k]ⁿ satisfieskⁿ/4|A|3kⁿ/4 then there are at leastk^n–1 edges betweenA and its complement.Our result is apparently the first example of an isoperimetric inequality for which the extremal sets do not form a nested family.We also give a best possible upper bound for the number of edges spanned by a subset of [k]ⁿ of given cardinality. In particular, forr=1,...,k we show that ifA[k]ⁿ satisfies |A|rⁿ then the subgraph of [k]ⁿ induced byA has average degree at most 2n(1–1/r).Research partially supported by NSF Grant DMS-8806097