On Some Brunn-Minkowski Inequalities for Probability Measures Under Symmetry

on YouTube ↗53m

" summary

Proposes a local normal flow approach that reduces Brunn–Minkowski–type concavity to bounding f''/f' for small Minkowski deformations, uses the Kolesnikov–Milman Rayleigh identity and integration-by-parts on the boundary, and converts the problem to energy estimates (notably torsional rigidity) by testing with 1 − ||x||_K^2. Proves sharp local concavity values for round k‑cylinders (equality cases) and gives lower bounds on the concavity power that blow up as the Gaussian mass of K tends to 1.

AI-written summary · about

" chapters (18)

1. 0:00Intro
2. 0:24Notation
3. 1:28Lebesgue isoperimetry
4. 3:11Gaussian isoperimetry
5. 4:56Connections to concentration of measure (small sample)
6. 6:01The local versions of Brunn-Minkowski type inequalities
7. 13:40A tighter question
8. 15:10An equivalence to an estimate with functions An equivalence to the function estimate
9. 19:05A version of a sharp bound
10. 20:41Case of round k-cylinders
11. 22:32Natural questions
12. 23:26A few words about energy minimization
13. 27:37Curious questions about energy minimization and such Question 1
14. 30:29A few words about the equality case characterization
15. 34:27Proof of the quality cases in BL via Hörmander's trick with boundary
16. 37:06Proof of the equality case characterization
17. 39:32Results for other measures
18. 44:41Isoperimetry with symmetry?