Newton’s lemma XXVIII on integrable ovals in higher dimensions and reflection groups

vassiliev, v. a;

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 R2, this is Lemma XXVIII from Newton’s
‘Philosophiae naturalis principia mathematica’.