{"ID":2839450,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2511.15232","arxiv_id":"2511.15232","title":"Optimal sets for the quantitative isoperimetric inequality in the plane with the barycentric distance","abstract":"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.","short_abstract":"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...","url_abs":"https://arxiv.org/abs/2511.15232","url_pdf":"https://arxiv.org/pdf/2511.15232v1","authors":"[\"Gisella Croce\",\"Antoine Henrot\"]","published":"2025-11-19T08:40:49Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"math.MG\"]","methods":"[]","has_code":false}