Quadratic Stability of the Brunn-Minkowski Inequality

Watch a mathematics seminar presentation exploring the quadratic stability of the Brunn-Minkowski inequality, a fundamental theorem in convex geometry. Delve into the relationship between volume control of subset sums in ℝn and examine how for sets A,B⊂ℝn of equal volume and parameter t∈(0,1), the inequality |tA+(1−t)B|≥|A| holds with equality only when A=B is convex. Learn about the progression from Ruzsa's early work suggesting linear stability to the proof of the quadratic stability conjecture, which demonstrates that for |tA+(1−t)B|≤(1+δ)|A|, we have |A△B|=On,t(δ√)|A| up to translation. Explore the implications of this proof for other geometric inequalities including the Prekopa-Leindler and Borell-Brascamb-Lieb inequalities in this collaborative work presented by Peter van Hintum from the Institute for Advanced Study, alongside colleagues Alessio Figalli and Marius Tiba.