Counting Regions with Bounded Surface Area

Define a cubical complex to be a collection of integer-aligned unit cubes in d dimensions. Lebowitz and Mazel (1998) proved that there are between and complexes containing a fixed cube with connected boundary of (d − 1)-volume n. In this paper we narrow these bounds to between and . We also show that there are connected complexes containing a fixed cube with (not necessarily connected) boundary of volume n.