Sharp Isoperimetric Inequalities for Affine Quermassintegrals

TL;DR

This paper proves Lutwak's conjecture that ellipsoids uniquely minimize affine quermassintegrals among convex bodies of fixed volume, extending classical inequalities and introducing new geometric tools.

Contribution

It confirms Lutwak's conjecture for all k, characterizes equality cases, and introduces the Projection Rolodex for affine geometric analysis.

Findings

Ellipsoids are the only local minimizers of affine quermassintegrals.

The work extends classical inequalities to a unified framework.

Introduces the Projection Rolodex and interprets Petty's inequality as a generalized Blaschke--Santaló inequality.

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-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 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--Santal\'o 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.