There are no algebraically integrable ovals in even-dimensional spaces
  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". 
Victor A. Vassiliev
math.AG