Linear Stability of the Brunn-Minkowski Inequality

Explore a nearly 2-hour mathematics seminar presentation delving into the linear stability of the Brunn-Minkowski inequality in convex geometry. Learn about fundamental concepts controlling volume subsets in ℝn, examining how for sets A,B⊂ℝn of equal volume and parameter t∈(0,1), |tA+(1−t)B|≥|A| holds true with equality when A=B is convex. Discover early work by Ruzsa and special cases suggesting linear stability results, where |tA+(1−t)B|≤(1+δ)|A| implies |co(A)∖A|=On,t(δ)|A|. Investigate the connection between these conjectures and discrete additive combinatorics, particularly focusing on geometric instances of the Polynomial Freiman-Ruzsa conjecture. Follow the detailed proof of the linear conjecture presented through collaborative research with Alessio Figalli and Marius Tiba.