Year: 2025
Type: article
Abstract: 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.
Source: Journal d Analyse Mathématique
Author: Mark Agranovsky
Institutions: Bar-Ilan University, Holon Institute of Technology
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W4416223863