Sharp quantitative stability for isoperimetric inequalities with homogeneous weights
Eleonora Cinti, Federico Glaudo, Aldo Pratelli, Xavier Ros‐Oton, João C. Serra
Abstract
We prove the sharp quantitative stability for a wide class of weighted isoperimetric inequalities. More precisely, we consider isoperimetric inequalities in convex cones with homogeneous weights. Inspired by the proof of such isoperimetric inequalities through the ABP method (see X. Cabré, X. Ros-Oton, and J. Serra [J. Eur. Math. Soc. (JEMS) 18 (2016), pp. 2971–2998]), we construct a new convex coupling (i.e., a map that is the gradient of a convex function) between a generic set<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"><mml:semantics><mml:mi>E</mml:mi><mml:annotation encoding="application/x-tex">E</mml:annotation></mml:semantics></mml:math></inline-formula>and the minimizer of the inequality (as in Gromov’s proof of the isoperimetric inequality). Even if this map does not come from optimal transport, and even if there is a weight in the inequality, we adapt the methods of [Figalli, Maggi, and Pratelli [Invent. Math. 182 (2010), pp. 167–211] and prove that if<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"><mml:semantics><mml:mi>E</mml:mi><mml:annotation encoding="application/x-tex">E</mml:annotation></mml:semantics></mml:math></inline-formula>is almost optimal for the inequality then it is quantitatively close to a minimizer<italic>up to translations</italic>. Then, a delicate analysis is necessary to rule out the possibility of translations. As a step of our proof, we establish a sharp regularity result for<italic>restricted</italic>convex envelopes of a function that might be of independent interest.