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Asymptotics of Perimeter-Minimizing Partitions

  • Quinn Maurmann (a1), Max Engelstein (a2), Anthony Marcuccio (a3) and Taryn Pritchard (a3)
Abstract

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.

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References
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[1] Bernstein, F., Über die isoperimetrische Eigenschaft des Kreises auf der Kugeloberfläche und in der Ebene. Math. Ann. 60(1905), no. 1, 117136. doi:10.1007/BF01447496
[2] Engelstein, M., The least-perimeter partition of a sphere into four equal areas. Discrete Comput. Geom. (2010), published online. doi:10.1007/s00454-009-9197-8
[3] Hales, T. C., The honeycomb conjecture. Discrete Comput. Geom. 25(2001), no. 1, 122.
[4] Hales, T. C., The honeycomb conjecture on the sphere. arxiv.org/abs/math/0211234
[5] Masters, J. D., The perimeter-minimizing enclosure of two areas in S 2 . Real Anal. Exchange 22(1996/97), no. 2, 645654.
[6] Morgan, F., Geometric Measure Theory. Fourth ed., Academic Press Inc., San Diego, CA, 2009.
[7] Morgan, F., Soap bubbles in R 2 and in surfaces. Pacific J. Math. 165(1994), no. 2, 347361.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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