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Slicing inequalities for measures of convex bodies
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abstract
We consider a generalization of the hyperplane problem to arbitrary measures in place of volume and to sections of lower dimensions. We prove this generalization for unconditional convex bodies and for duals of bodies with bounded volume ratio. We also prove it for arbitrary symmetric convex bodies under the condition that the dimension of sections is less than $\lambda n$ for some $\lambda\in (0,1).$ The constant depends only on $\lambda.$ Finally, we show that the behavior of the minimal sections for some measures may be different from the case of volume.
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Isomorphic Busemann--Petty for arbitrary measures: the sharp order
The optimal constant in the isomorphic Busemann-Petty problem for arbitrary even densities has the sharp order √n: a new lower bound C_n ≥ c√n matches the known upper bound C_n ≤ √n.
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