Estimates for moments of general measures on convex bodies

For p ≥ 1, n ∈ ℕ, and an origin-symmetric convex body K in ℝ ⁿ, let (formula presented) be the outer volume ratio distance from K to the class L ⁿ _p of the unit balls of n-dimensional subspaces of L _p. 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 ⊆ ℝ ⁿ 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 ℝ ⁿ. 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.