Slicing inequalities for subspaces of $L_p$

Abstract:

We prove slicing inequalities for measures of the unit balls of subspaces of $L_p$, $-\infty <p<\infty .$ For example, for every $k\in \mathbb {N}$ there exists a constant $C(k)$ such that for every $n\in \mathbb {N},\ k<n$, every convex $k$-intersection body (unit ball of a normed subspace of $L_{-k})$ $L$ in $\mathbb {R}^n$ and every measure $\mu$ with non-negative even continuous density in $\mathbb {R}^n,$ \[ \mu (L)\ \le \ C(k) \max _{\xi \in S^{n-1}} \mu (L\cap \xi ^\bot )\ |L|^{1/n} \ ,\] where $\xi ^\bot$ is the central hyperplane in $\mathbb {R}^n$ perpendicular to $\xi ,$ and $|L|$ is the volume of $L.$ This and other results are in the spirit of the hyperplane problem of Bourgain. The proofs are based on stability inequalities for intersection bodies and estimates for the Banach-Mazur distance from the unit ball of a subspace of $L_p$ to the class of intersection bodies.