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 E 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 E is almost optimal for the inequality then it is quantitatively close to a minimizer up to translations. 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 restricted convex envelopes of a function that might be of independent interest.

Cinti E., Glaudo F., Pratelli A., Ros-Oton X., Serra J. (2022). SHARP QUANTITATIVE STABILITY FOR ISOPERIMETRIC INEQUALITIES WITH HOMOGENEOUS WEIGHTS. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 375(3), 1509-1550 [10.1090/tran/8525].

SHARP QUANTITATIVE STABILITY FOR ISOPERIMETRIC INEQUALITIES WITH HOMOGENEOUS WEIGHTS

Cinti E.;
2022

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 E 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 E is almost optimal for the inequality then it is quantitatively close to a minimizer up to translations. 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 restricted convex envelopes of a function that might be of independent interest.
2022
Cinti E., Glaudo F., Pratelli A., Ros-Oton X., Serra J. (2022). SHARP QUANTITATIVE STABILITY FOR ISOPERIMETRIC INEQUALITIES WITH HOMOGENEOUS WEIGHTS. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 375(3), 1509-1550 [10.1090/tran/8525].
Cinti E.; Glaudo F.; Pratelli A.; Ros-Oton X.; Serra J.
File in questo prodotto:
File Dimensione Formato  
abp_stability.pdf

accesso aperto

Tipo: Postprint
Licenza: Licenza per accesso libero gratuito
Dimensione 697.49 kB
Formato Adobe PDF
697.49 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/864340
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 9
  • ???jsp.display-item.citation.isi??? 9
social impact