Discrete Isoperimetric Problems

It is shown that there exists a linear ordering of the points in $I_ + ^n $ or $I^n $ (Cartesian products of nonnegative integers or integers) such that the first j points in this ordering is a configuration that minimizes the number of boundary points (points not in the set that have Euclidean distance one from the set) among sets of size j. The relation of the $I_ + ^n $ result to Macaulay’s theorem is deduced also.