On the average volume of sections of convex bodies

The average section functional $\mathrm{a}\mathrm{s}(K)$ of a centered convex body in ${\mathbb{R}}^n$ is the average volume of central hyperplane sections of $K$: \begin{equation*}{\rm as}(K)=\int_{S^{n-1}}|K\cap \xi^{\perp }|,d\sigma (\xi ).\end{equation*} We study the question if there exists an absolute constant $C>0$ such that for every $n$, for every centered convex body $K$ in ${\mathbb R}^n$andforevery0<k<n,$${\rm as}(K)\ls C^k|K|^{\frac{k}{n}},\max_{E\in {\rm Gr}{n-k}}{\rm as}(K\cap E).$$Weobservethatthecase$k=1$ is equivalent to the hyperplane conjecture. We show that this inequality holds true in full generality if one replaces $C$ by $C{L}_K$ or $Cd{\rm ovr}(K,{\cal{BP}}k^n)$,where$L_K$istheisotropicconstantof$K$and$d{\rm ovr}(K,{\cal{BP}}_k^n)$istheoutervolumeratiodistancefrom$K$ to the class ${\mathcal{B}\mathcal{P}}_k^n$ of generalized $k$-intersection bodies. We also compare $\mathrm{a}\mathrm{s}(K)$ to the average of $\mathrm{a}\mathrm{s}(K\cap E)$ over all $k$-codimensional sections of $K$. We examine separately the dependence of the constants on the dimension in the case where $K$ is in some of the classical positions as well as the natural lower dimensional analogue of the average section functional.