Abstract
The 1956 Busemann-Petty problem asks whether symmetric convex bodies in ℝn with larger central hyperplane sections must also have greater volume. The solution to the problem has recently been completed, and the answer is negative ifn≥5 and affirmative whenn≤4. We show a more general result, where the inequalities for the volume of central sections are replaced by similar inequalities for the derivatives of the parallel section functions at zero. The dimension of affirmative answer goes up together with the order of the derivatives. The proof is based on a version of Parseval's formula.
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Part of this work was done when the author was visiting the Weizmann Institute of Science. Research supported in part by the NSF Grant DMS-9531594.
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Koldobsky, A. A generalization of the Busemann-Petty problem on sections of convex bodies. Isr. J. Math. 110, 75–91 (1999). https://doi.org/10.1007/BF02808176
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DOI: https://doi.org/10.1007/BF02808176