Calculus of Variations and Geometric Measure Theory
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F. Dayrens - S. Masnou - M. Novaga - M. Pozzetta

Connected perimeter of planar sets

created by novaga on 21 Jun 2019
modified on 08 Apr 2020


Accepted Paper

Inserted: 21 jun 2019
Last Updated: 8 apr 2020

Journal: Adv. Calc. Var.
Year: 2019

ArXiv: 1906.09814 PDF


We introduce a notion of connected perimeter for planar sets defined as the lower semicontinuous envelope of perimeters of approximating sets which are measure-theoretically connected. A companion notion of simply connected perimeter is also studied.

We prove a representation formula which links the connected perimeter, the classical perimeter, and the length of suitable Steiner trees. We also discuss the application of this notion to the existence of solutions to a nonlocal minimization problem.


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