Convex bodies with algebraic section volume functions

TL;DR

This paper demonstrates that among smooth, strictly convex bodies, ellipsoids can be uniquely characterized by algebraic equations involving their section volume functions, linking geometric shape to algebraic properties.

Contribution

It proves that ellipsoids are uniquely determined by algebraic equations of their section volume functions among smooth convex bodies.

Findings

Ellipsoids are characterized by algebraic equations of their section volume functions.

The result generalizes previous work on polynomially integrable domains.

The study connects geometric shape detection with algebraic properties of section 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.