Geodesic
Sharp isoperimetric inequalities via the ABP method
Journal of the European Mathematical Society, Tome 18 (2016) no. 12, pp. 2971-2998

Voir la notice de l'article provenant de la source EMS Press

Given an arbitrary convex cone of Rn, we find a geometric class of homogeneous weights for which balls centered at the origin and intersected with the cone are minimizers of the weighted isoperimetric problem in the convex cone. This leads to isoperimetric inequalities with the optimal constant that were unknown even for a sector of the plane. Our result applies to all nonnegative homogeneous weights in Rn satisfying a concavity condition in the cone. The condition is equivalent to a natural curvature-dimension bound and also to the nonnegativeness of a Bakry-Emery Ricci tensor. Even that our weights are nonradial, still balls are minimizers of the weighted isoperimetric problem. A particular important case is that of monomial weights. Our proof uses the ABP method applied to an appropriate linear Neumann problem.
DOI : 10.4171/jems/659
Classification : 28-XX, 35-XX, 49-XX
Keywords: Isoperimetric inequalities, densities, convex cones, homogeneous weights, Wulff shapes, ABP method 
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     author = {Xavier Cabr\'e and Xavier Ros-Oton and Joaquim Serra},
     title = {Sharp isoperimetric inequalities via the {ABP} method},
     journal = {Journal of the European Mathematical Society},
     pages = {2971--2998},
     publisher = {mathdoc},
     volume = {18},
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     year = {2016},
     doi = {10.4171/jems/659},
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Xavier Cabré; Xavier Ros-Oton; Joaquim Serra. Sharp isoperimetric inequalities via the ABP method. Journal of the European Mathematical Society, Tome 18 (2016) no. 12, pp. 2971-2998. doi: 10.4171/jems/659

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