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arXiv · 2511.15232

Optimal sets for the quantitative isoperimetric inequality in the plane with the barycentric distance

Gisella Croce·Antoine Henrot
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Abstract

In a recent paper, C. Gambicchia and A. Pratelli proved a quantitative isoperimetric inequality involving the isoperimetric deficit $\delta(K)$ and the barycentric distance $\lambda_0(K)$ for sets $K\subset \mathbb{R}^N$ with given diameter $D$ and measure. In this work we are interested in the optimal sets for this inequality in the plane, i.e. sets that minimize the ratio $\delta(K)/\lambda_0(K)^2$. We prove existence of optimal sets (at least when $D$ is large enough), regularity and express the optimality conditions. Moreover, we prove that the optimal sets have exactly two connected components and their boundary does not contain any arc of circle.

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Gisella Croce, Antoine Henrot. 2025-11-19. Optimal sets for the quantitative isoperimetric inequality in the plane with the barycentric distance. https://arxiv.org/abs/2511.15232

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