Abstract
The affine quermassintegrals associated to a convex body in $\R^n$ are affine-invariant analogues of the classical intrinsic volumes from the Brunn--Minkowski theory, and thus constitute a central 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 volume, the $k$-th affine quermassintegral is minimized precisely on the family of ellipsoids. The known cases $k=1$ and $k=n-1$ correspond to the classical 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 single unified framework. In fact, it turns out that ellipsoids are the only \emph{local} minimizers with respect to the Hausdorff topology. In addition, 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 Yehudayoff.