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SUMMARY:Emanuel Milman (Technion\, Haifa)
DTSTART:20201208T153000Z
DTEND:20201208T163000Z
DTSTAMP:20260423T035815Z
UID:OAGAS/47
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OAGAS/47/">S
 harp Isoperimetric Inequalities for Affine Quermassintegrals</a>\nby Emanu
 el Milman (Technion\, Haifa) as part of Online asymptotic geometric analys
 is seminar\n\n\nAbstract\nThe affine quermassintegrals associated to a con
 vex body in $\\R^n$ are affine-invariant analogues of the classical intrin
 sic volumes from the Brunn--Minkowski theory\, and thus constitute a centr
 al pillar of affine convex geometry. They were introduced in the 1980's by
  E. Lutwak\, who conjectured that among all convex bodies of a given volum
 e\, the $k$-th affine quermassintegral is minimized precisely on the famil
 y of ellipsoids. The known cases $k=1$ and $k=n-1$ correspond to the class
 ical Blaschke--Santal\\'o and Petty projection inequalities\, respectively
 . In this work we confirm Lutwak's conjecture\, including characterization
  of the equality cases\, for all values of $k=1\,\\ldots\,n-1$\, in a sing
 le unified framework. In fact\, it turns out that ellipsoids are the only 
 \\emph{local} minimizers with respect to the Hausdorff topology. In additi
 on\, we address a related conjecture of Lutwak on the validity of certain 
 Alexandrov--Fenchel-type inequalities for affine (and more generally $L^p$
 -moment) quermassintegrals. The case $p=0$ corresponds to a sharp averaged
  Loomis--Whitney isoperimetric inequality. Based on joint work with Amir Y
 ehudayoff.\n
LOCATION:https://researchseminars.org/talk/OAGAS/47/
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