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

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Abstract

Given a non-increasing and radially symmetric kernel in \(L ^ 1 _{\textrm{loc}} (\mathbb {R}^ 2 ; \mathbb {R}_+)\), 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 \(N \ge 3\), 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 \(N\ge 5\) 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.

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Notes

  1. The construction is taken from the MathWorld page https://mathworld.wolfram.com/GrahamsBiggestLittleHexagon.html.

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Authors and Affiliations

  1. Centre de Mathématiques Appliquées, CNRS, École Polytechnique, Institut Polytechnique de Paris, 91120, Palaiseau, France

    Beniamin Bogosel

  2. Université Savoie Mont Blanc, CNRS UMR 5127, Laboratoire de Mathématiques, Campus Scientifique, 73376, Le-Bourget-Du-Lac, France

    Dorin Bucur

  3. Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133, Milan, Italy

    Ilaria Fragalà

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  1. Beniamin Bogosel
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Correspondence to Dorin Bucur.

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Bogosel, B., Bucur, D. & Fragalà, I. The nonlocal isoperimetric problem for polygons: Hardy–Littlewood and Riesz inequalities. Math. Ann. 389, 1835–1882 (2024). https://doi.org/10.1007/s00208-023-02683-x

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