Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
Cancel
×
×
Home
  • Home
Article
  • Access

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.

    •
      • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Asymptotics of Perimeter-Minimizing Partitions
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Asymptotics of Perimeter-Minimizing Partitions
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Asymptotics of Perimeter-Minimizing Partitions
      Available formats
      ×
Copyright
COPYRIGHT: © Canadian Mathematical Society 2010
References
Hide All
[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/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, 1–22.
[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, 645–654.
[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, 347–361.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

CAPTCHA *

Skip to the audio challenge
×
MathJax

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed

Cancel
Confirm
×