A comparison theorem for the planar strip isoperimetric problem

Abstract

We consider the isoperimetric problem with planar density equal to 1 on a horizontal strip and to a constant $\lambda >1$ outside the strip. We resolve a conjecture made by Cañete-Miranda Jr-Vittone by showing that every four-arc region admits a three-arc competitor with the same weighted area and strictly smaller weighted perimeter. The argument applies to every $\lambda >1$ and gives an explicit positive lower bound for the perimeter difference.

7 pages