Stability and slicing inequalities for intersection bodies

We prove a generalization of the hyperplane inequality for intersection bodies, where volume is replaced by an arbitrary measure $\mu$ with even continuous density and sections are of arbitrary dimension $n-k,1\le k<n\mathrm{.}$ If $K$ is a generalized $k$-intersection body, then $$\mu (K)\le \frac{n}{n-k}{c}_{n,k}\underset{H}{\max }\mu (K\cap H){\text{vol}}_n(K{)}^{k\mathrm{/}n}\mathrm{.}$$ Here $c_{n,k}=\mathrm{\mid }{B}_2^n{\mathrm{\mid }}^{(n-k)\mathrm{/}n}\mathrm{/}\mathrm{\mid }{B}_2^{n-k}\mathrm{\mid }<1,$ $\mathrm{\mid }{B}_2^n\mathrm{\mid }$ is the volume of the unit Euclidean ball, and maximum is taken over all $(n-k)$-dimensional subspaces of ${\mathbb{R}}^n\mathrm{.}$ The constant is optimal, and for each intersection body the inequality holds for every $k\mathrm{.}$ We also prove a stronger “difference” inequality. The proof is based on stability in the lower dimensional Busemann-Petty problem for arbitrary measures in the following sense. Let $\text{e}>0,1\le k<n\mathrm{.}$ Suppose that $K$ and $L$ are origin-symmetric star bodies in ${\mathbb{R}}^n,$ and $K$ is a generalized $k$-intersection body. If for every $(n-k)$-dimensional subspace $H$ of ${\mathbb{R}}^n$ $$\mu (K\cap H)\le \mu (L\cap H)+\text{e},$$ then $$\mu (K)\le \mu (L)+\frac{n}{n-k}{c}_{n,k}{\text{vol}}_n(K{)}^{k\mathrm{/}n}\text{e}\mathrm{.}$$