# Connected perimeter of planar sets

## TL;DR

This paper introduces a new concept of connected perimeter for planar sets, linking it to classical perimeter and Steiner trees, with applications to nonlocal minimization problems.

## Contribution

It defines a connected perimeter as a lower semi-continuous envelope and establishes a representation formula connecting it to classical perimeter and Steiner trees.

## Key findings

- Connected perimeter is characterized via a representation formula.
- The notion is applied to prove existence results for nonlocal minimization problems.
- A related simply connected perimeter is also studied.

## Abstract

We introduce a notion of connected perimeter for planar sets defined as the lower semi-continuous 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.