A remark on perimeter–diameter and perimeter–circumradius inequalities under lattice constraints

Abstract

In this note, we are concerned with planar convex sets whose interior is free from points of the integer lattice. Such lattice-free convex sets have been extensively studied with respect to determining sharp relations between any two of the geometric functionals: perimeter, diameter, circumradius, inradius, minimal width and area. The cases perimeter–diameter and perimeter–circumradius have been left open and are the subject of our investigations. We formulate precise conjectures on linear inequalities for these settings and prove them in special yet illustrative cases. Moreover, we obtain sharp non-linear inequalities that hold unconditionally.