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

By Alessio Figalli, E. Indrei. 2012

77 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7372cb3243f769a45657f6afe3ae1a80f54aa093

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Symmetry results for critical anisotropic p-Laplacian equations in convex cones (2019, background): “Sobolev inequalities have been studied for more general norms as well as in convex cones (see [5, 9, 20, 21, 28, 29]), where they take the form ‖u‖Lp∗ (Σ) ≤ SΣ,H‖H(∇u)‖Lp(Σ) , (1.”
Complemented Brunn–Minkowski inequalities and isoperimetry for homogeneous and non-homogeneous measures (2013, methodology): “The uniqueness of general isoperimetric minimizers in the Lions–Pacella scenario in the case that Σ\{0} is smooth was obtained by Lions and Pacella themselves [21], and using a different approach by Ritoré and Rosales [26]; the smoothness assumption has been recently removed by Figalli and Indrei [1”
The quantitative isoperimetric inequality and related topics (2015, background): “Paper [63] is devoted to the stability of the relative isoperimetric inequality in a cone while [11] contains a characterization of the equality cases in the perimeter inequality for Steiner symmetrization in any codimension, as well as a stability result for this inequality for a class of convex se”
Sharp isoperimetric inequalities via the ABP method (2013, unknown): “This fact has also been used recently by Figalli and Indrei [22] to prove a quantitative isoperimetric inequality in convex cones.”
Quantitative logarithmic Sobolev inequalities and stability estimates (2014, result): “Examples of such results are the recent quantitative stability estimates for Sobolev [12, 19], Brunn-Minkowski [17, 16], and isoperimetric inequalities [20, 18, 15, 23].”