Stability and slicing inequalities for intersection bodies

Abstract

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 $n-k,\ 1\le k <n.$ If $K$ is a generalized $k$-intersection body, then $$\mu(K)\,\leq\,\frac{n}{n-k}c_{n,k}\max_{H} \mu(K\cap H) \vol_n(K)^{k/n}.$$ Here $c_{n,k} = |B_2^n|^{(n-k)/n}/|B_2^{n-k}|<1,$ $|B_2^n|$ is the volume of the unit Euclidean ball, and maximum is taken over all $(n-k)$-dimensional subspaces of $\R^n.$ The constant is optimal, and for each intersection body the inequality holds for every $k.$ 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 $\e>0,\ 1\le k <n.$ Suppose that $K$ and $L$ are origin-symmetric star bodies in $\R^n,$ and $K$ is a generalized $k$-intersection body. If for every $(n-k)$-dimensional subspace $H$ of $\R^n$ $$\mu(K\cap H)\leq \mu(L\cap H)+\e,$$ then $$\mu(K)\leq \mu(L) +\frac{n}{n-k}c_{n,k} \vol_n(K)^{k/n}\e.$$