Lutwak's affine quermassintegral inequality for real, complex, and quaternionic convex bodies

Let $\mathbb{F}$ be one of $\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$, let $p=\dim_{\mathbb{R}}\mathbb{F}$, and let $K\subset \mathbb{F}^n$ be a convex body with non-empty interior. Write $|K|$ for its volume, let $\operatorname{Gr}_m(n,\mathbb{F})$ be the Grassmannian of $m$-dimensional $\mathbb{F}$-subspaces of $\mathbb{F}^n$ with invariant probability measure $dE$, and for $E\in\operatorname{Gr}_m(n,\mathbb{F})$ let $P_EK$ denote the orthogonal projection of $K$ onto $E$ and $|P_EK|$ its volume in $E$. Let $\kappa_{rp}$ denote the volume of the Euclidean unit ball in $\mathbb{R}^{rp}$. Lutwak's affine quermassintegral conjecture.

$$|K|^{-m} \geq \frac{(\kappa_{mp})^n}{(\kappa_{np})^m} \int_{\operatorname{Gr}_m(n,\mathbb{F})} |P_EK|^{-n}\,dE$$

with equality if and only if $K$ is a real, complex, or quaternionic ellipsoid, according as $\mathbb{F}=\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$. This conjecture was recently confirmed by Milman--Yehudayoff, including the complex and quaternionic cases.