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A Sharp Stability Result for the Relative Isoperimetric Inequality Inside Convex Cones

2011; Springer Science+Business Media; Volume: 23; Issue: 2 Linguagem: Inglês

10.1007/s12220-011-9270-4

1559-002X

Alessio Figalli, Emanuel Indrei,

Geometric Analysis and Curvature Flows

The relative isoperimetric inequality inside an open, convex cone $\mathcal{C}$ states that, at fixed volume, $B_{r} \cap\mathcal{C}$ minimizes the perimeter inside $\mathcal{C}$ . Starting from the observation that this result can be recovered as a corollary of the anisotropic isoperimetric inequality, we exploit a variant of Gromov's proof of the classical isoperimetric inequality to prove a sharp stability result for the relative isoperimetric inequality inside $\mathcal{C}$ . Our proof follows the line of reasoning in Figalli et al.: Invent. Math. 182:167–211 (2010), though several new ideas are needed in order to deal with the lack of translation invariance in our problem.