Borell-Brascamp-Lieb inequalities: rigidity and stability

La tesi è dedicata allo studio delle cosiddette disuguaglianze di Borell-Brascamp-Lieb, note in letteratura come forme funzionali della disuguaglianza di Brunn-Minkowski. L'intento della tesi è duplice: da una parte si prefigge come manuale dettagliato delle disuguaglianze di Borell-Brascamp-Lieb, affrontando varie estensioni e proprietà più o meno note in letteratura; in secondo luogo si concentra sulla questione della stabilità di tali disuguaglianze, citando i risultati più significativi ed esibendo i contributi originali ottenuti, tratti dagli articoli: 1) A. Rossi, P. Salani, Stability for Borell-Brascamp-Lieb inequalities, Geometric Aspects of Functional Analysis - Israel Seminar (GAFA) 2014-2016 (B. Klartag and E. Milman Eds), Springer Lecture Notes in Mathematics 2169 (2017); 2) A. Rossi, P. Salani, Stability for a strengthened one-dimensional Borell-Brascamp-Lieb inequality, Applicable Analysis (2018). All the Borell-Brascamp-Lieb inequalities can be read as the functional counterparts of the celebrated Brunn-Minkowski inequality, and they have been widely studied in the last decades. The thesis focuses on two main targets. The first is to produce a complete and detailed overview on the results (old and new) on the Borell-Brascamp-Lieb inequalities, the second is to investigate some open questions on the quantitative version of such inequalities. The thesis is divided in 7 chapters. The first five contain the overview on the state of the art, classical and alternative proofs of both Borell-Brascamp-Lieb and Brunn-Minkowski inequalities, theequality cases and some stability results. Chapter 6 and Chapter 7 are devoted to describe the original contributions of the author in the field. Precisely in Chapter 6 a strengthened version of the one dimensional Borell-Brascamp-Liebinequality is proved, while in Chapter 7 the goal is to prove a general quantitative versions of the Borell-Brascamp-Lieb inequalities without concavity assumptions on the involved function. The original results are contained in the following two papers: • A. Rossi, P. Salani, Stability for Borell-Brascamp-Lieb inequalities, Geometric Aspects of Functional Analysis - Israel Seminar (GAFA) 2014-2016 (B. Klartag - E. Milman Eds), Springer Lecture Notes in Mathematics 2169 (2017); • A. Rossi, P. Salani, Stability for a strengthened one-dimensional Borell-Brascamp- Lieb inequality, Applicable Analysis (2018).