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<p>The section volume function <i>A</i><sub><i>K</i></sub>(<i>ξ, t</i>), <i>ξ</i> ∈ ℝ<sup><i>n</i></sup>, <i>t</i> ∈ ℝ, of a body <i>K</i> ⊂ ℝ<sup><i>n</i></sup> evaluates the (<i>n</i> − 1)-dimensional volume of the cross-section of <i>K</i> by the hyperplane {<i>x·ξ</i> = <i>t</i>}. We are concerned with the question: can the shape of a body <i>K</i> be detected from an algebraic type of its section function? We prove that among strictly convex bodies <i>K</i> with <i>C</i><sup><i>∞</i></sup> boundaries, ellipsoids are completely described by the algebraic equation <i>qA</i><Stack> <sub><i>K</i></sub> <sup><i>m</i></sup> </Stack> + <i>p</i> = 0, where <i>m</i> ∈ ℕ and <i>q</i> = <i>q</i>(<i>ξ</i>), <i>p</i> = <i>p</i>(<i>ξ, t</i>) 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 ([<CitationRef CitationID="CR15">15</CitationRef>], [<CitationRef CitationID="CR1">1</CitationRef>]), [<CitationRef CitationID="CR3">3</CitationRef>]) on polynomially integrable domains.</p>

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Convex bodies with algebraic section volume functions

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年11月12日

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  • Mark Agranovsky

摘要

The section volume function AK(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.