Riesz-type Inequalities and Overdetermined Problems for Triangles and Quadrilaterals

We consider Riesz-type nonlocal interaction energies over polygons. We prove the analog of the Riesz inequality in this discrete setting for triangles and quadrilaterals, and obtain that among all N-gons with fixed area, the nonlocal energy is maximized by a regular polygon, for N= 3, 4. Further we derive necessary first-order stationarity conditions for a polygon with respect to a restricted class of variations, which will then be used to characterize regular N-gons, for N= 3, 4, as solutions to an overdetermined free boundary problem.