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

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.