Algebraically integrable bodies and related properties of the Radon transform

Abstract

Generalizing Lemma 28 from Newton’s “Principia” [25], Arnold [10] asked for a complete characterization of algebraically integrable domains. In this chapter we describe the current state of Arnold’s problems. We also consider closely related problems involving the Radon transform of indicator functions.

Acknowledgement

The third named author was supported in part by the U. S. National Science Foundation Grant DMS-2054068. The fifth author was supported in part by NSERC. This material is partially based on the work supported by the U. S. National Science Foundation grant DMS-1929284. The third and fifth authors were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Harmonic Analysis and Convexity semester program. The main part of the work of the fourth author was done at the Steklov Mathematical Institute, Moscow.

1 Introduction

The questions considered in this survey belong to the area of geometric tomography (see the book [17]), which lies at the crossroads between convex geometry and integral geometry and can be defined as the study of geometric properties of solids based on data about their sections and projections.

We study algebraic properties of two important volumetric characteristics in geometric tomography. For a body (compact set with non-empty interior) K in Rn, ξ∈Sn−1, and t∈R, the cutoff functions of K represent the n-dimensional volume of the parts of K cut by the hyperplane perpendicular to ξ at distance t from the origin:

The section function of K is the (n−1)-dimensional volume of the section of K by the same hyperplane:

Here R stands for the Radon transform, χK is the indicator (characteristic function) of K, ⟨x,ξ⟩ is the inner (scalar) product in Rn, and dx is the Lebesgue measure on Rn or {x:⟨x,ξ⟩=t}, correspondingly. Clearly, the cutoff functions and the section function are related via differentiation in t.

Most of our problems take root in Lemma 28 about ovals from Newton’s Principia [25]; see also the discussion in [8], [9], [34]. Newton proved that if K is a convex infinitely smooth domain in R2, then the cutoff function of K cannot appear as the solution of a polynomial equation involving the parameters of the cutting hyperplane. Formalizing the question and extending it to higher dimensions, Arnold [10] asked whether there exist domains with smooth boundaries in Rn (apart from ellipsoids for odd n) for which the cutoff functions VK± are branches of an algebraic function. Recall that a function f(ξ,t) is algebraic if there exists a non-zero polynomial Φ(ξ,t,w) of n+2 variables such that

In Section 2, we present the current state of Arnold’s problems. In particular, it was proved in [29] that there are no algebraically integrable bodies with infinitely smooth boundaries in even dimensions. However, the odd-dimensional case is still open.

In Sections 3–7, we consider similar questions that are motivated by Arnold’s problem and address the single-valued section function AK(ξ,t) rather than the multi-valued cutoff function VK(ξ,t). Therefore, we study geometric properties of bodies K from the point of view of algebraic properties of their Radon transform AK. The following definition is similar to Definition 1.1.

The essential difference between the two definitions is that in Definition 1.2 we do not assume that Ψ is a polynomial in ξ, as we do in Definition 1.1, so the section function AK(ξ,t) is algebraic only with respect to the variable t.

Note that if K is algebraically integrable (i. e., the cutoff function VK± is algebraic), then the section function AK(ξ,t) is also algebraic as the derivative

of an algebraic function. Thus, the class of domains with algebraic Radon transform contains algebraically integrable domains.

Our basic example is the unit ball Bn in Rn. In this case

If n is odd, then ABn(ξ,t) is a polynomial in t. Applying an affine transformation to Bn we obtain that AK(ξ,t) is a polynomial in t if n is odd and K is an ellipsoid.

In this chapter, we consider classes of bodies K satisfying Definition 1.2 with the defining polynomial Ψ of a certain form. The property of ellipsoids in odd-dimensional spaces mentioned above gives rise to the following.

In the case of polynomially integrable domains, the equation Ψ(ξ,t,w)=0 in Definition 1.2 has the form...