Differentiation properties of the perimeter-to-area ratio for finitely many overlapped unit squares

Abstract

In this paper we examine finite unions of unit squares in same plane and consider the ratio of perimeter to area of these unions. In 1998, T. Keleti published the conjecture that this ratio never exceeds 4. Here we study the continuity and differentiability of functions derived from the geometry of the union of those squares. Specifically we show that if there is a counterexample to Keleti’s conjecture, there is also one where the associated ratio function is differentiable.