New affine inequalities and projection mean ellipsoids
The authors introduce the notion of the projection mean ellipsoid $P_jK$ of a convex body $K\subset\mathbb{R}^n$, $1\leq j\leq n-1$. This is the unique origin-symmetric ellipsoid $E$ which maximizes volume subject to a constraint $\bar{\Lambda}_j(K,E)\leq 1$, the latter being related to the $j$th projection function $V_j(K|\cdot)$ of $K$. With this notion, the authors establish an inequality for the affine quermassintegrals \[ \Lambda_j(K)=n\frac{\omega_n}{\omega_j}\left(\int_{G_{n,j}}V_j(K|\xi)^{-n}\mathrm{d}\mu_j(\xi)\right)^{-1/n}, \] with $\mu_j$ being the normalized Haar measure on the Grassmannian $G_{n,j}$, $j\in\{1,\ldots,n-1\}$. Namely, \[ \Lambda_j(K)^n\geq n^n\omega_n^{n-j}V_n(P_jK)^j \] for $K\subset\mathbb{R}^n$ a convex body and $j\in\{1,\ldots,n-1\}$. (For $j\in\{2,\ldots,n-1\}$, equality holds precisely for the ellipsoid, and for $j=1$, equality holds iff $K$ has an $SL(n)$ image of constant width.) Furthermore, the authors show that \[ V_n(K^\ast)V_n(P_1K)\leq\omega_n^2 \] for $K=-K\subset\mathbb{R}^n$ a convex body, with equality precisely for the ellipsoid.
scientific article; zbMATH DE number 7023785
scientific article