An analogue of polynomially integrable bodies in even-dimensional spaces

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

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Abstract

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
Volume529
Issue number2
DOIs
StatePublished - 15 Jan 2024

Bibliographical note

Publisher Copyright:
© 2023 Elsevier Inc.

Keywords

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

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