Convex bodies with algebraic section volume functions

The section volume function A_K(ξ, t), ξ ∈ ℝⁿ, t ∈ ℝ, of a body K ⊂ ℝⁿ 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.