# The nonlocal isoperimetric problem for polygons: Hardy-Littlewood and Riesz inequalities

## TL;DR

This paper explores nonlocal isoperimetric problems for polygons, establishing optimality of regular polygons for Hardy-Littlewood inequalities and revealing complex behaviors for Riesz inequalities depending on the number of sides and kernel choice.

## Contribution

It extends classical inequalities to nonlocal polygonal domains, proving regular polygons are optimal for Hardy-Littlewood and analyzing symmetry breaking in Riesz inequalities.

## Key findings

- Regular N-gons are optimal for Hardy-Littlewood inequalities.
- Optimality of regular polygons for Riesz inequalities depends on N and kernel parameters.
- Symmetry breaking can occur for N ≥ 5 in Riesz inequalities.

## Abstract

Given a non-increasing and radially symmetric kernel in $L ^ 1 _{\rm loc} (\Bbb{R} ^ 2 ; \Bbb{R}_+)$, we investigate counterparts of the classical Hardy-Littlewood and Riesz inequalities when the class of admissible domains is the family of polygons with given area and $N$ sides. The latter corresponds to study the polygonal isoperimetric problem in nonlocal version. We prove that, for every $N \geq 3$, the regular $N$-gon is optimal for Hardy-Littlewood inequality. Things go differently for Riesz inequality: while for $N = 3$ and $N = 4$ it is known that the regular triangle and the square are optimal, for $N\geq 5$ we prove that symmetry or symmetry breaking may occur (i.e. the regular $N$-gon may be optimal or not), depending on the value of $N$ and on the choice of the kernel.