# Minimizers for nonlocal perimeters of Minkowski type

## TL;DR

This paper investigates a nonlocal perimeter functional inspired by Minkowski content, exploring minimizers, inequalities, and constructions that preserve image features while reducing noise.

## Contribution

It introduces new results on existence, rigidity, classification, and construction of minimizers for a nonlocal Minkowski-type perimeter functional.

## Key findings

- Existence and classification of minimizers
- Isoperimetric and Poincaré inequalities established
- Construction of planelike minimizers under perturbations

## Abstract

We study a nonlocal perimeter functional inspired by the Minkowski content, whose main feature is that it interpolates between the classical perimeter and the volume functional. This problem is related by a generalized coarea formula to a Dirichlet energy functional in which the energy density is the local oscillation of a function. These two nonlocal functionals arise in concrete applications, since the nonlocal character of the problems and the different behaviors of the energy at different scales allow the preservation of details and irregularities of the image in the process of removing white noises, thus improving the quality of the image without losing relevant features. In this paper, we provide a series of results concerning existence, rigidity and classification of minimizers, compactness results, isoperimetric inequalities, Poincar\'e-Wirtinger inequalities and density estimates. Furthermore, we provide the construction of planelike minimizers for this generalized perimeter under a small and periodic volume perturbation.