Sharp isoperimetric inequalities for affine quermassintegrals

Abstract:

The affine quermassintegrals associated to a convex body in $\mathbb {R}^n$ 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–Santaló 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 local minimizers with respect to the Hausdorff topology.

For the proof, we introduce a number of new ingredients, including a novel construction of the Projection Rolodex of a convex body. In particular, from this new view point, Petty’s inequality is interpreted as an integrated form of a generalized Blaschke–Santaló inequality for a new family of polar bodies encoded by the Projection Rolodex. We extend these results to more general $L^p$-moment quermassintegrals, and interpret the case $p=0$ as a sharp averaged Loomis–Whitney isoperimetric inequality.