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arXiv · 1702.00429

On polynomially integrable convex bodies

Alexander Koldobsky·Alexander Merkurjev·Vladyslav Yaskin
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Abstract

An infinitely smooth convex body in $\mathbb R^n$ is called polynomially integrable of degree $N$ if its parallel section functions are polynomials of degree $N$. We prove that the only smooth convex bodies with this property in odd dimensions are ellipsoids, if $N\ge n-1$. This is in contrast with the case of even dimensions and the case of odd dimensions with $N<n-1$, where such bodies do not exist, as it was recently shown by Agranovsky.

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Alexander Koldobsky, Alexander Merkurjev, Vladyslav Yaskin. 2017-02-01. On polynomially integrable convex bodies. https://arxiv.org/abs/1702.00429

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