Optimal sets for the quantitative isoperimetric inequality in the plane with the barycentric distance
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911276403261440 |
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| author | Croce, Gisella Henrot, Antoine |
| author_facet | Croce, Gisella Henrot, Antoine |
| contents | In a recent paper, C. Gambicchia and A. Pratelli proved a quantitative isoperimetric inequality involving the isoperimetric deficit $δ(K)$ and the barycentric distance $λ_0(K)$ 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 $δ(K)/λ_0(K)^2$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_15232 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal sets for the quantitative isoperimetric inequality in the plane with the barycentric distance Croce, Gisella Henrot, Antoine Optimization and Control Metric Geometry 28A75, 49J45, 49J53, 49Q10, 49Q20 In a recent paper, C. Gambicchia and A. Pratelli proved a quantitative isoperimetric inequality involving the isoperimetric deficit $δ(K)$ and the barycentric distance $λ_0(K)$ 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 $δ(K)/λ_0(K)^2$. 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. |
| title | Optimal sets for the quantitative isoperimetric inequality in the plane with the barycentric distance |
| topic | Optimization and Control Metric Geometry 28A75, 49J45, 49J53, 49Q10, 49Q20 |
| url | https://arxiv.org/abs/2511.15232 |