Convex bodies with algebraic section volume functions

Abstract

The section volume function $A_K(\xi,t), \ \xi \in \mathbb R^n, \ t \in \mathbb R,$ of a body $K \subset \mathbb R^n$ evaluates the $(n-1)$-dimensional volume of the cross-section $K$ by the hyperplane $\{ x \cdot \xi=t \}.$ 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 $C^{\infty}$ boundaries, ellipsoids are completely described by the algebraic equation $qA_K^m+p=0,$ where $m \in \mathbb N$ and $q=q(\xi), \ p=p(\xi,t)$ 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.