The isomorphic Busemann–Petty problem for s-concave measures

This paper considers the isomorphic lower-dimensional Busemann–Petty problem. Fix \(k\in \{1,2,\ldots ,n-1\}\). If \(\mu \) is a measure on \(\mathbb {R}^n\) with an even non-negative density, and K, L are origin-symmetric convex bodies in \(\mathbb {R}^n\) such that for every \((n-k)\)-dimensional subspaces H of \(\mathbb {R}^n\), $$\begin{aligned}\mu (K\cap H)\le \mu (L\cap H),\end{aligned}$$ does there exist a constant \(\mathcal {L}\) such that $$\begin{aligned}\mu (K)\le {\mathcal {L}}^\frac{nk}{n-k}\mu (L)?\end{aligned}$$ We provide different new estimates for the constant \(\mathcal {L}\) for arbitrary s-concave measures, containing the cases \(0

Keywords:

• Isomorphism
• Mathematical analysis
• Pure mathematics
• Mathematics
• Algebraic geometry
• Hyperbolic geometry
• Combinatorics
• Differential geometry
• Regular polygon
• Projective geometry
• Linear subspace

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