For p ≥ 1, n ∈ ℕ, and an origin-symmetric convex body K in ℝn, let (formula presented) be the outer volume ratio distance from K to the class Ln p of the unit balls of n-dimensional subspaces of Lp. We prove that there exists an absolute constant c > 0 such that (formula presented) This result follows from a new slicing inequality for arbitrary measures, in the spirit of the slicing problem of Bourgain. Namely, there exists an absolute constant C > 0 so that for any p ≥ 1, any n ∈ ℕ, any compact set K ⊆ ℝn of positive volume,∫ and any Borel measurable function f ≥ 0 on K, (formula presented) where the supremum is taken over all affine hyperplanes H in ℝn. Combining the above display with a recent counterexample for the slicing problem with arbitrary measures from the work of the second and third authors [J. Funct. Anal. 274 (2018), pp. 2089–2112], we get the lower estimate from the first display. In turn, the second inequality follows from an estimate for the p-th absolute moments of the function f (formula presented) Finally, we prove a result of the Busemann-Petty type for these moments.
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Received by the editors December 16, 2017, and, in revised form, February 1, 2018. 2010 Mathematics Subject Classification. Primary 52A20; Secondary 46B07. This material is based upon work supported by the U. S. National Science Foundation under Grant DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester. The first-and third-named authors were supported in part by the NSF Grants DMS-1612961 and DMS-1700036. The second-named author was supported in part by a European Research Council (ERC) grant.