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- W2079483320 abstract "We prove a generalization of the hyperplane inequality for intersection bodies, where volume is replaced by an arbitrary measure μ with even continuous density and sections are of arbitrary dimension n − k, 1 ≤ k < n. If K is a generalized k-intersection body, then $$mu(K),leq,frac{n}{n-k}c_{n,k} max_{H} mu(Kcap H) {rm Vol}_n(K)^{k/n}.$$ 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 $${varepsilon >0 , 1le k < n}$$ . 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 $${mathbb{R}^{n}}$$ $$mu(Kcap H)leq mu(Lcap H)+varepsilon,$$ then $$mu(K)leq mu(L) +frac{n}{n-k}c_{n,k} {rm Vol}_n(K)^{k/n} varepsilon.$$" @default.
- W2079483320 created "2016-06-24" @default.
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- W2079483320 date "2012-04-25" @default.
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- W2079483320 title "Stability and slicing inequalities for intersection bodies" @default.
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- W2079483320 doi "https://doi.org/10.1007/s10711-012-9729-x" @default.
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