Pick’s Theorem in Two-Dimensional Subspace ofR3

Abstract

In the Euclidean spaceR3, denote the set of all points with integer coordinate byZ3. For any two-dimensional simple lattice polygonP, we establish the following analogy version of Pick’s Theorem,kIP+1/2BP-1, whereBPis the number of lattice points on the boundary ofPinZ3,IPis the number of lattice points in the interior ofPinZ3, andkis a constant only related to the two-dimensional subspace includingP.