Estimates for moments of general measures on convex bodies

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We prove several estimates for the moments of arbitrary measures on convex bodies. We apply these estimates to show a new slicing inequality for measures on convex bodies. We also deduce estimates for the outer volume ratio distance from an arbitrary centrally-symmetric convex body in Rⁿ to the class of unit balls of n-dimensional subspaces of Lₚ-spaces. Finally, we prove a result of the Busemann-Petty type for these moments.