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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
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Abstract |
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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
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Mathematical Subject Classification 2010
Primary: 26B05
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Milestones
Received: 2 October 2014
Revised: 19 November 2014
Accepted: 20 November 2014
Published: 28 September 2015
Communicated by Frank Morgan
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Authors |
| Paul D. Humke | |
| Department of Mathematics St. Olaf College 1520 St. Olaf Avenue Northfield, MN 55057 United States |
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| Cameron Marcott | |
| Department of Mathematics St. Olaf College 1520 St. Olaf Avenue Northfield, MN 55057 United States |
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| Bjorn Mellem | |
| Department of Mathematics St. Olaf College 1520 St. Olaf Avenue Northfield, MN 55057 United States |
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| Cole Stiegler | |
| Department of Mathematics St. Olaf College 1520 St. Olaf Avenue Northfield, MN 55057 United States |
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