An analogue of polynomially integrable bodies in even-dimensional spaces

M. Agranovsky, A. Koldobsky, D. Ryabogin, V. Yaskin

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A bounded domain K⊂Rn is called polynomially integrable if the (n−1)-dimensional volume of the intersection K with a hyperplane Π polynomially depends on the distance from Π to the origin. It was proved in [7] that there are no such domains with smooth boundary if n is even, and if n is odd then the only polynomially integrable domains with smooth boundary are ellipsoids. In this article, we modify the notion of polynomial integrability for even n and consider bodies for which the sectional volume function is a polynomial up to a factor which is the square root of a quadratic polynomial, or, equivalently, the Hilbert transform of this function is a polynomial. We prove that ellipsoids in even dimensions are the only convex infinitely smooth bodies satisfying this property.

Original languageEnglish
Article number127071
JournalJournal of Mathematical Analysis and Applications
Issue number2
StatePublished - 15 Jan 2024

Bibliographical note

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  • Ellipsoids
  • Hilbert transform
  • Polynomials
  • Radon transform
  • Volumes


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