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 $\hat{I}¼$ 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 $$μ(K)\,\leq\,\frac{n}{n-k}c_{n,k}\max_{H} μ(K\cap H) \vol_n(K)^{k/n}.$$ 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 ${\mathbb{R}}^n.$ 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 ${\mathbb{R}}^n,$ and $K$ is a generalized $k$-intersection body. If for every $(n-k)$-dimensional subspace $H$ of ${\mathbb{R}}^n$ $$μ(K\cap H)\leq μ(L\cap H)+\e,$$ then $$μ(K)\leq μ(L) +\frac{n}{n-k}c_{n,k} \vol_n(K)^{k/n}\e.$$