Abstract
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.
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Communicated by Marco Abate.
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Figalli, A., Indrei, E. A Sharp Stability Result for the Relative Isoperimetric Inequality Inside Convex Cones. J Geom Anal 23, 938–969 (2013). https://doi.org/10.1007/s12220-011-9270-4
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DOI: https://doi.org/10.1007/s12220-011-9270-4