Two problems concerning the area-perimeter ratio of lattice-point-free regions in the plane (Q1922672)

The authors consider the integer lattice in the euclidean plane and prove that for a lattice-point-free simple connection (i.e. not necessarily convex) region \(M\) with area \(A(M)\) and perimeter \(P(M)\) holds: \(A(M)<\gamma P(M)\), where \(\gamma=0.58\dots\) is the solution of a transcendental equation. In particular the inequality is tight. As a consequence they prove a tight Dido-type for simple connected regions.