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Asymptotics of Perimeter-Minimizing Partitions

Published online by Cambridge University Press:  20 November 2018

Quinn Maurmann ,
Max Engelstein ,
Anthony Marcuccio  and
Taryn Pritchard
Quinn Maurmann
Affiliation:
Department of Mathematics, UCLA, Los Angeles, CA, U.S.A. e-mail: qmaurmann@math.ucla.edu
Max Engelstein
Affiliation:
Department of Mathematics, Yale University, New Haven, CT, U.S.A.e-mail: max.engelstein@yale.edu
Anthony Marcuccio
Affiliation:
Department of Mathematics and Statistics, Williams College, Williamstown, MA, U.S.A. e-mail: 08anm@williams.edu, 08tbp@williams.edu
Taryn Pritchard
Affiliation:
Department of Mathematics and Statistics, Williams College, Williamstown, MA, U.S.A. e-mail: 08anm@williams.edu, 08tbp@williams.edu
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Abstract

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We prove that the least perimeter $P(n)$ of a partition of a smooth, compact Riemannian surface into $n$ regions of equal area $A$ is asymptotic to $n/2$ times the perimeter of a planar regular hexagon of area $A$. Along the way, we derive tighter estimates for flat tori, Klein bottles, truncated cylinders, and Möbius bands.

Keywords

53C42

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 53 , Issue 3 , 01 September 2010 , pp. 516 - 525
Copyright
Copyright © Canadian Mathematical Society 2010

References

[1] Bernstein, F., Über die isoperimetrische Eigenschaft des Kreises auf der Kugeloberfläche und in der Ebene. Math. Ann. 60(1905), no. 1, 117–136. doi:10.1007/BF01447496Google Scholar
[2] Engelstein, M., The least-perimeter partition of a sphere into four equal areas. Discrete Comput. Geom. (2010), published online. doi:10.1007/s00454-009-9197-8Google Scholar
[3] Hales, T. C., The honeycomb conjecture. Discrete Comput. Geom. 25(2001), no. 1, 1–22.Google Scholar
[4] Hales, T. C., The honeycomb conjecture on the sphere. arxiv.org/abs/math/0211234Google Scholar
[5] Masters, J. D., The perimeter-minimizing enclosure of two areas in S 2 . Real Anal. Exchange 22(1996/97), no. 2, 645–654.Google Scholar
[6] Morgan, F., Geometric Measure Theory. Fourth ed., Academic Press Inc., San Diego, CA, 2009.Google Scholar
[7] Morgan, F., Soap bubbles in R 2 and in surfaces. Pacific J. Math. 165(1994), no. 2, 347–361.Google Scholar