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Extensions of the Busemann-Petty Problem for Arbitrary Measures

The classical Busemann-Petty problem asks whether smaller central hyperplane sections of origin-symmetric convex bodies necessarily imply smaller total volume. Zvavitch studied this question for arbitrary measures with continuous even densities, providing sufficient conditions for affirmative cases in terms of the distributional behavior of the ratio between the densities involved. We refine this result by extending it to a broader class of distributions and allowing a distinct pair of densities for each body, one for hyperplane sections and another for the full volume. We also present some examples illustrating cases not covered by previous results.