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

Given a non-increasing and radially symmetric kernel in L_loc¹(R²;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≥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≥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.