Pick’s Theorem in Two-Dimensional Subspace of R3

Abstract EN

In the Euclidean space R3, denote the set of all points with integer coordinate by Z3.

For any two-dimensional simple lattice polygon P, we establish the following analogy version of Pick’s Theorem, kIP+1/2BP-1, where BP is the number of lattice points on the boundary of P in Z3, IP is the number of lattice points in the interior of P in Z3, and k is a constant only related to the two-dimensional subspace including P.