Double Bubbles on the Real Line with Log-Convex Density

Eliot Bongiovanni 1 , Leonardo Di Giosia 2 , Alejandro Diaz 3 , Jahangir Habib 4 , Arjun Kakkar 4 , Lea Kenigsberg 5 , Dylanger Pittman 4 , Nat Sothanaphan 6 ,  and Weitao Zhu 4
  • 1 Michigan State , University, USA
  • 2 Rice University, Houston, , Texas, USA
  • 3 University of Maryland, , College Park, USA
  • 4 Williams College, , Williamstown, USA
  • 5 Columbia University in the City of , New York, USA
  • 6 Massachusetts Institute of Technology, , Cambridge, USA

Abstract

The classic double bubble theorem says that the least-perimeter way to enclose and separate two prescribed volumes in ℝN is the standard double bubble. We seek the optimal double bubble in ℝN with density, which we assume to be strictly log-convex. For N = 1 we show that the solution is sometimes two contiguous intervals and sometimes three contiguous intervals. In higher dimensions we think that the solution is sometimes a standard double bubble and sometimes concentric spheres (e.g. for one volume small and the other large).

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Analysis and Geometry in Metric Spaces (AGMS) is a fully peer-reviewed, open access electronic journal that publishes cutting-edge original research on analytical and geometrical problems in metric spaces and their mathematical applications. It features articles making connections among relevant topics in this field.

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