2023/02/22 by Bogosel, Beniamin, Bucur, Dorin, Fragalà, Ilaria · 1 citation
#FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.2302.11677
Given a non-increasing and radially symmetric kernel in L ^ 1 \rm loc (ℝ ^ 2 ; 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.