  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.
