# Functional and geometric inequalities

### Published/accepted papers

- A quantitative stability result for the Prékopa-Leindler inequality for arbitrary measurable functions (with K. J. Böröczky and J. P. G. Ramos)
*Ann. Inst. H. Poincaré Anal. Non Linéaire*, to appear - Sharp gradient stability for the Sobolev inequality (with Y. Ru-Ya Zhang)

*Duke Math. J.*171 (2022), no. 12, 2407–2459. - 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. - A sharp Freiman type estimate for semisums in two and three dimensional Euclidean spaces (with D. Jerison)
*Ann. Sci. Éc. Norm. Supér.*, to appear - 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. - On the sharp stability of critical points of the Sobolev inequality (with F. Glaudo)
*Arch. Ration. Mech. Anal.*237 (2020), no. 1, 201–258. - 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. - The sharp quantitative Euclidean concentration inequality (with F. Maggi and C. Mooney)
*Camb. J. Math.*6 (2018), no. 1, 59-87. - Gradient stability for the Sobolev inequality: the case p≥2 (with R. Neumayer)

*J. Eur. Math. Soc. (JEMS)*21 (2019), no. 2, 319-354. - Quantitative stability for the Brunn-Minkowski inequality (with D. Jerison)
*Adv. Math.*314 (2017), 1-47. - Rigidity and stability of Caffarelli's log-concave perturbation theorem (with G. De Philippis)

*Nonlinear Anal.*154 (2017), 59-70. - 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. - Characterization of isoperimetric sets inside almost-convex cones (with E. Baer)

*Discrete Contin. Dyn. Syst.*37 (2017), no. 1, 1-14. - Quantitative stability for sumsets in R
^{n}(with D. Jerison)

*J. Eur. Math. Soc. (JEMS)*17 (2015), no. 5, 1079-1106. - 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. - On the isoperimetric problem for radial log-convex densities (with F. Maggi)

*Calc. Var. Partial Differential Equations*48 (2013), no. 3-4, 447-489. - Sharp stability theorems for the anisotropic Sobolev and log-Sobolev inequalities on functions of bounded variation (with F. Maggi and A. Pratelli)

*Adv. Math.*242 (2013), 80-101. - A sharp stability result for the relative isoperimetric inequality inside convex cones (with E. Indrei)

*J. Geom. Anal.*23 (2013), no. 2, 938-969. - Isoperimetric-type inequalities on constant curvature manifolds (with Y. Ge)

*Adv. Calc. Var.*5 (2012), no. 3, 251-284. - A mass transportation approach to quantitative isoperimetric inequalities (with F. Maggi and A. Pratelli)

*Invent. Math.*182 (2010), no. 1, 167-211. - 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. - A note on Cheeger sets (with F. Maggi and A. Pratelli)

*Proc. Amer. Math. Soc.*137 (2009), no.6, 2057-2062. - A geometric lower bound on Grad's number

*ESAIM Control Optim. Calc. Var.*15 (2009), no. 3, 569-575.

### Submitted papers

- Sharp quantitative stability of the Brunn-Minkowski inequality (with P. van Hintum and M. Tiba)
- Strong stability of convexity with respect to the perimeter via a quantitative Alexandrov theorem with optimal decay (with Y. Ru-Ya Zhang)
- Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence (with J. Dolbeault, M. J. Esteban, R. L. Frank, M. Loss)

### Surveys and lecture notes

- Stability results for the Brunn-Minkowski inequality

Colloquium De Giorgi 2013 and 2014, 119-127,*Colloquia*5, Ed. Norm., Pisa, 2014. - Quantitative stability results for the Brunn-Minkowski inequality

Proceedings of the ICM 2014. - Stability in geometric and functional inequalities

Proceedings of the 6th European Congress of Mathematics, 2012. - 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. - Optimal transport, Euler equations, Mather and DiPerna-Lions theories.

Mémoire d'Habilitation à Diriger de Recherche (HDR). Nice, 2009.