On Minimal Perimeter Polyminoes

This paper explores proofs of the isoperimetric inequality for 4-connected shapes on the integer grid ℤ2, and its geometric meaning. Pictorially, we discuss ways to place a maximal number unit square tiles on a chess board so that the shape they form has a minimal number of unit square neighbors. Previous works have shown that “digital spheres” have a minimum of neighbors for their area. We here characterize all shapes that are optimal and show that they are all close to being digital spheres. In addition, we show a similar result when the 8-connectivity metric is assumed (i.e. connectivity through vertices or edges, instead of edge connectivity as in 4-connectivity).