# Functional and geometric inequalities

### Published/accepted papers

1. Sharp gradient stability for the Sobolev inequality (with Y. Ru-Ya Zhang)
Duke Math. J. 171 (2022), no. 12, 2407–2459.
2. Strong stability for the Wulff inequality with a crystalline norm (with Y. Ru-Ya Zhang)
Comm. Pure Appl. Math. 75 (2022), no. 2, 422–446.
3. A sharp Freiman type estimate for semisums in two and three dimensional Euclidean spaces (with D. Jerison)
Ann. Sci. Éc. Norm. Supér., to appear
4. Symmetry results for critical anisotropic p-Laplacian equations in convex cones (with G. Ciraolo and A. Roncoroni)
Geom. Funct. Anal. 30 (2020), no. 3, 770-803.
5. On the sharp stability of critical points of the Sobolev inequality (with F. Glaudo)
Arch. Ration. Mech. Anal. 237 (2020), no. 1, 201–258.
6. A quantitative analysis of metrics on R^n with almost constant positive scalar curvature, with applications to fast diffusion flows (with G. Ciraolo and F. Maggi)
Int. Math. Res. Not. IMRN 2018, no. 21, 6780-6797.
7. The sharp quantitative Euclidean concentration inequality (with F. Maggi and C. Mooney)
Camb. J. Math. 6 (2018), no. 1, 59-87.
8. Gradient stability for the Sobolev inequality: the case p≥2 (with R. Neumayer)
J. Eur. Math. Soc. (JEMS) 21 (2019), no. 2, 319-354.
9. Quantitative stability for the Brunn-Minkowski inequality (with D. Jerison)
10. Rigidity and stability of Caffarelli's log-concave perturbation theorem (with G. De Philippis)
Nonlinear Anal. 154 (2017), 59-70.
11. Quantitative stability of the Brunn-Minkowski inequality for sets of equal volume (with D. Jerison)
Chin. Ann. Math. Ser. B 38 (2017), no. 2, 393-412.
12. Characterization of isoperimetric sets inside almost-convex cones (with E. Baer)
Discrete Contin. Dyn. Syst. 37 (2017), no. 1, 1-14.
13. Quantitative stability for sumsets in Rn (with D. Jerison)
J. Eur. Math. Soc. (JEMS) 17 (2015), no. 5, 1079-1106.
14. A geometric approach to correlation inequalities in the plane (with F. Maggi and A. Pratelli)
Ann. Inst. H. Poincaré Probab. Stat. 50 (2014), no. 1, 1-14.
15. On the isoperimetric problem for radial log-convex densities (with F. Maggi)
Calc. Var. Partial Differential Equations 48 (2013), no. 3-4, 447-489.
16. Sharp stability theorems for the anisotropic Sobolev and log-Sobolev inequalities on functions of bounded variation (with F. Maggi and A. Pratelli)
17. A sharp stability result for the relative isoperimetric inequality inside convex cones (with E. Indrei)
J. Geom. Anal. 23 (2013), no. 2, 938-969.
18. Isoperimetric-type inequalities on constant curvature manifolds (with Y. Ge)
Adv. Calc. Var. 5 (2012), no. 3, 251-284.
19. A mass transportation approach to quantitative isoperimetric inequalities (with F. Maggi and A. Pratelli)
Invent. Math. 182 (2010), no. 1, 167-211.
20. A refined Brunn-Minkowski inequality for convex sets (with F. Maggi and A. Pratelli)
Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 6, 2511-2519.
21. A note on Cheeger sets (with F. Maggi and A. Pratelli)
Proc. Amer. Math. Soc. 137 (2009), no.6, 2057-2062.
22. A geometric lower bound on Grad's number
ESAIM Control Optim. Calc. Var. 15 (2009), no. 3, 569-575.

### Submitted papers

1. A quantitative stability result for the Prékopa-Leindler inequality for arbitrary measurable functions (with K. J. Böröczky and J. P. G. Ramos)

### Lecture notes and reviews

1. Stability results for the Brunn-Minkowski inequality
Colloquium De Giorgi 2013 and 2014, 119-127, Colloquia 5, Ed. Norm., Pisa, 2014.
2. Quantitative stability results for the Brunn-Minkowski inequality
Proceedings of the ICM 2014.
3. Stability in geometric and functional inequalities
Proceedings of the 6th European Congress of Mathematics, 2012.
4. Quantitative isoperimetric inequalities, with applications to the stability of liquid drops and crystals
Concentration, functional inequalities and isoperimetry, 77-87, Contemp. Math. 545, Amer. Math. Soc., Providence, RI, 2011.
5. Optimal transport, Euler equations, Mather and DiPerna-Lions theories.
Mémoire d'Habilitation à Diriger de Recherche (HDR). Nice, 2009.