Vol. 8, No. 5, 2015

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

Paul D. Humke, Cameron Marcott, Bjorn Mellem and Cole Stiegler

Vol. 8 (2015), No. 5, 875–891
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.

Keywords
Keleti, perimeter-to-area ratio
Mathematical Subject Classification 2010
Primary: 26B05
Milestones
Received: 2 October 2014
Revised: 19 November 2014
Accepted: 20 November 2014
Published: 28 September 2015

Communicated by Frank Morgan
Authors
Paul D. Humke
Department of Mathematics
St. Olaf College
1520 St. Olaf Avenue
Northfield, MN 55057
United States
Cameron Marcott
Department of Mathematics
St. Olaf College
1520 St. Olaf Avenue
Northfield, MN 55057
United States
Bjorn Mellem
Department of Mathematics
St. Olaf College
1520 St. Olaf Avenue
Northfield, MN 55057
United States
Cole Stiegler
Department of Mathematics
St. Olaf College
1520 St. Olaf Avenue
Northfield, MN 55057
United States