On convex lattice polygons

Let II be a convex lattice polygon with b boundary points and c (5 1) interior points. We show that for any given a , the number b satisfies b 5 2e + 7 , and identify the polygons for which equality holds. A lattice polygon II is a simple polygon whose vertices are points of the integral lattice. We let A = 4(11) denote the area of II , b{U) the number of lattice points on the boundary of II , and e(II) the number of lattice points interior to II. A(n) = %2)(n) + c(n)-1. Nosarzewska [7] and more recently Wills L41, have established inequalities relating the area, perimeter, and number of interior points of a convex lattice polygon. It is our purpose here to establish a simple necessary condition for II to be convex. We set /(II) = Z>(II)-2c(II). Using Pick's formula we can obtain alternative expressions for f(T[) : %/(H) = b{Ti)-A{Jl)-1 and = 4(n)-Lattice polygons which can be obtained from one another using integral unimodular transformations or translations are said to be equivalent. The property of convexity, and the quantities A, b, a , and f are easily seen to be invariant under equivalence.