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arXiv · 1108.2631

Stability and slicing inequalities for intersection bodies

Alexander Koldobsky·Dan Ma
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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 $n-k,\ 1\le k 0,\ 1\le k <n.$ 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 $$\mu(K)\leq \mu(L) +\frac{n}{n-k}c_{n,k} \vol_n(K)^{k/n}\e.$$

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Alexander Koldobsky, Dan Ma. 2011-08-12. Stability and slicing inequalities for intersection bodies. https://arxiv.org/abs/1108.2631

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