Multidimensional Newton lemma on algebraically integrable oval

Lemma XXVIII from Newton's “Principia” states that there is no convex bounded domain with infinitely smooth boundary in the plane such that the areas cut off from the domain by affine lines form an algebraic function on the space of lines. In the talk, the same fact for arbitrary bounded domains with smooth boundaries in any even-dimensional spaces will be proved. This is a solution to the V. I. Arnold's problem on the multidimensional generalization of Newton lemma, posed in 1987 in connection with the 300th anniversary of “Principia”. The proof is based on Picard-Lefschetz theory and theory of reflection groups.