There are no algebraically integrable ovals in even-dimensional spaces

TL;DR

This paper proves that in even-dimensional Euclidean spaces, no smooth bounded domains exist where the volume cut off by hyperplanes depends algebraically on the hyperplanes, extending a classical result from Newton's Principia.

Contribution

The paper establishes a non-existence theorem for algebraically integrable domains in even-dimensional spaces, generalizing Newton's lemma to higher dimensions.

Findings

No algebraically integrable bounded domains in even-dimensional Euclidean spaces.

Extends classical results from 2D to higher even dimensions.

Provides a rigorous proof of the non-existence in the smooth boundary case.

Abstract

We prove that there are no bounded domains with smooth boundaries in even-dimensional Euclidean spaces, such that the volumes cut off from them by affine hyperplanes depend algebraically on these hyperplanes. For convex ovals in $R^2$, this is Lemma XXVIII from Newton's "Principia".