Stability and slicing inequalities for intersection bodies

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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 μ(K)≤n/n-kcₙ,ₖₕ μ(K∩ H) ₙ(K)ᵏ/ⁿ. Here cₙ,ₖ = |B₂ⁿ|⁽ⁿ⁻ᵏ⁾/ⁿ/|B₂ⁿ⁻ᵏ|<1, |B₂ⁿ| is the volume of the unit Euclidean ball, and maximum is taken over all (n-k)-dimensional subspaces of 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 >0,1≤ k <n. Suppose that K and L are origin-symmetric star bodies in ⁿ, and K is a generalized k-intersection body. If for every (n-k)-dimensional subspace H of ⁿ μ(K∩ H)≤ μ(L∩ H)+, then μ(K)≤ μ(L) +n/n-kcₙ,ₖ ₙ(K)ᵏ/ⁿ.