2024/09/28 by Mark Agranovsky, Agranovsky, Mark
Mathematics · #44A12 #51M25 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2409.19373
openalex publication_date 2024/09/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The section volume function AK(ξ,t), ξ∈ \mathbb Rn, t ∈ \mathbb R, of a body K ⊂ \mathbb Rn evaluates the (n-1)-dimensional volume of the cross-section 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 qAKm+p=0, where m ∈ \mathbb 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 on polynomially integrable domains.