An analog of polynomially integrable bodies in even-dimensional spaces
| Autor: | Agranovsky, Mark, Koldobsky, Alexander, Ryabogin, Dmitry, Yaskin, Vladyslav |
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| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | A bounded domain $K \subset \mathbb R^n$ is called polynomially integrable if the $(n-1)$-dimensional volume of the intersection $K$ with a hyperplane $\Pi$ polynomially depends on the distance from $\Pi$ 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. Comment: 10 pages |
| Databáze: | arXiv |
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