Sharp Isoperimetric Inequalities for Affine Quermassintegrals

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Proves sharp lower bounds for the affine quermassintegrals, resolves Lutwak's conjecture for all k with equality only for ellipsoids, by showing each affine quermassintegral never increases under Steiner symmetrization. Key tools are a new auxiliary construction called the projection rolodex (a collection of level sets living over lower-dimensional Grassmannians), a weighted Blaschke–Petkantschin identity defining a μ_u measure, and convexity of a joint section-volume under shadow‑system deformations.

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" chapters (14)

1. 0:00Introduction and setup
2. 1:07Quermass integrals
3. 4:26Isoparametric inequalities
4. 6:16Lutwak's conjecture
5. 8:43Rank one cases
6. 13:34Related results
7. 17:15Main results
8. 19:35Equality and minimality
9. 23:01Approaches and barriers
10. 27:29Projection rolodex
11. 34:18Proof sketch
12. 37:01Key convexity proposition
13. 48:23Extensions and Loomis-Whitney
14. 55:10Q&A and closing