On algebraically integrable domains in Euclidean spaces

TL;DR

This paper proves that in odd-dimensional Euclidean spaces, domains with algebraic volume functions cut off by hyperplanes are ellipsoids, addressing a question posed by V.I. Arnold about algebraically integrable domains.

Contribution

It establishes that the only algebraically integrable domains in odd dimensions are ellipsoids, providing a partial answer to Arnold's question.

Findings

Domains with algebraic hyperplane volume functions are ellipsoids in odd dimensions.

The result applies to domains with smooth boundaries and no real singularities in the volume function.

Addresses a longstanding question in integral geometry.

Abstract

Let $D$ be a bounded domain $D$ in $\mathbb R^n $ with infinitely smooth boundary and $n$ is odd. We prove that if the volume cut off from the domain by a hyperplane is an algebraic function of the hyperplane, free of real singular points, then the domain is an ellipsoid. This partially answers a question of V.I. Arnold: whether odd-dimensional ellipsoids are the only algebraically integrable domains?