Convex bodies with algebraic section volume functions

The section volume function A K ( ξ, t ), ξ ∈ ℝ n , t ∈ ℝ, of a body K ⊂ ℝ n evaluates the ( n − 1)-dimensional volume of the cross-section of K by the hyperplane { x·ξ = t }. We are concerned with the question: can the shape of a body K be detected from an algebraic type of its section function? We prove that among strictly convex bodies K with C ∞ boundaries, ellipsoids are completely described by the algebraic equation qA K m + p = 0, where m ∈ ℕ and q = q ( ξ ), p = p ( ξ, t ) are polynomials. The result is motivated by Arnold’s problem on algebraically integrable domains (which, in turn, has its roots in Newton’s Lemma about ovals), and generalizes known results ([15], [1]), [3]) on polynomially integrable domains.