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.

The Radon transform of the characteristic function of a domain in ℝ^n (the section function) evaluates the volumes of plane sections and is one of the basic metric characteristics in geometric tomography. We are interested in the following question: how is the geometry of a domain linked with the algebraic properties of the section function? The question is motivated by the Arnold problem (in turn, rooted in Newton’s Lemma About Ovals) on algebraically integrable domains and has been the subject of extensive study over the past decade. A brief overview of work related to the above question and recent new results on bodies with algebraic Radon transforms are presented.

A bounded domain K⊂Rn is called polynomially integrable if the (n−1)-dimensional volume of the intersection K with a hyperplane Π polynomially depends on the distance from Π to the origin. It was proved in [7] that there are no such domains with smooth boundary if n is even, and if n is odd then the only polynomially integrable domains with smooth boundary are ellipsoids. In this article, we modify the notion of polynomial integrability for even n and consider bodies for which the sectional volume function is a polynomial up to a factor which is the square root of a quadratic polynomial, or, equivalently, the Hilbert transform of this function is a polynomial. We prove that ellipsoids in even dimensions are the only convex infinitely smooth bodies satisfying this property.

The spherical means Radon transform Mf(x, r) is defined by the integral of a function f in Rnover the sphere S(x, r) of radius r centered at a x, normalized by the area of the sphere. The problem of reconstructing f from the data Mf(x, r) where x belongs to a hypersurface gamma subset of Rn and r is an element of (0, infinity) has important applications in modern imaging modalities, such as photo-and thermoacoustic tomography. When gamma coincides with the boundary & part;? of a bounded (convex) domain ? subset of R-n, a function supported within ? can be uniquely recovered from its spherical means known on gamma. We are interested in explicit inversion formulas for such a reconstruction. If gamma = & part;?, such formulas are only known for the case when gamma is an ellipsoid (or one of its partial cases). This gives rise to a question: can explicit inversion formulas be found for other closed hypersurfaces gamma? In this article we prove, for the so-called "universal backprojection inversion formulas', that their extension to non-ellipsoidal domains ? is impossible, and therefore ellipsoids constitute the largest class of closed convex hypersurfaces for which such formulas hold.

Generalizing Lemma 28 from Newton's ``Principia", Arnold asked for a complete characterization of algebraically integrable domains. In this paper we describe the current state of Arnold's problems. We also consider closely related problems about the Radon transform of indicator functions.

This article gives a brief overview of the research in microlocal analysis, tomography, and integral geometry of Professor Eric Todd Quinto, Robinson Professor of Mathematics at Tufts University, along with the collaborators and colleagues who influenced his work.

Koldobsky, Merkurjev and Yaskin proved in (Koldobsky in Adv Math 320:876-886, 2017) that given a convex body $$K \subset {\mathbb {R}}^n, \ n$$ is odd, with smooth boundary, such that the volume of the intersection $$K \cap L$$ of K with a hyperplane $$L \subset {\mathbb {R}}^n$$ (the sectional volume function) depends polynomially on the distance t of L to the origin, then the boundary of K is an ellipsoid. In even dimension, the sectional volume functions are never polynomials in t, nevertheless in the case of ellipsoids their squares are. We conjecture that the latter property fully characterizes ellipsoids and, disregarding the parity of the dimension, ellipsoids are the only convex bodies with smooth boundaries whose sectional volume functions are roots (of some power) of polynomials. In this article, we confirm this conjecture for planar domains, bounded by algebraic curves. A multidimensional version in terms of chords lengths, i.e., of X-ray transform of the characteristic function, is given. The result is motivated by Arnold’s conjecture on characterization of algebraically integrable bodies.

Let K be a compact convex body in ℝ^n. For any affine line L, denote χ_K(L)=∫_Lχ_K(x)dl(x), where dl is the arc length measure, the X-ray transform of the characteristic function χ_K, i.e., the length of the chord K ∩ L. We prove that if K is bounded by a C^∞ real algebraic hypersurface ∂ K and the X-ray transform χ_K(L) behaves, under small parallel translations of the line L to the distance t, as the m-th root of a polynomial of t, for some fixed m ∈ℕ, then ∂ K is an ellipsoid.

after a long illness that did nothing to weaken the memory and love that many of us felt for him.I was his first doctoral student, taking a several complex variables class with him in the fall of 1978, as a new graduate student at the University of Maryland.Carlos was the dream advisor for a young, but passionate, aspiring mathematician.As his beloved daughter Nadia recounted in her beautiful eulogy, he used to answer my naive questions with variations of "this is actually not trivial," something that encouraged my willingness to ask questions,

We study non-geodesic Funk-type transforms associated with cross-sections of the n-sphere by k-dimensional planes passing through an arbitrary fixed point inside the sphere. The main results include injectivity conditions for these transforms, inversion formulas, and connection with geodesic Funk transforms. We also show that, unlike the case of planes through a single common center, the integrals over spherical sections by planes through two distinct centers provide the corresponding reconstruction problem a unique solution.

We study Funk-type transforms associated with intersections of the unit sphere in Rn with lower-dimensional affine planes passing through a given point inside or outside the sphere. Our goal is to investigate injectivity of such "paired" transforms generated by two families of planes centered at distinct points. Necessary and sufficient conditions of injectivity are obtained in terms of geometry of the centers location. The technique used is based upon the action of the automorphism group of the unit ball and a related billiard-like dynamics on the sphere.

We study Funk-type transforms on the unit sphere in R^n associated with cross-sections of the sphere by lower-dimensional planes passing through an arbitrary fixed point inside the sphere or outside. Our main concern is injectivity of the corresponding paired transforms generated by two families of planes centered at distinct points. Necessary and sufficient conditions for the paired transforms to be injective are obtained, depending on geometrical configuration of the centers. Our method relies on the action of the automorphism group of the unit ball and the relevant billiard-like dynamics on the sphere.

Let K be a bounded body in R-n, n is odd, with infinitely smooth boundary. We prove that if the volume cut off from the body by a hyper-plane is a free of real singularities algebraic function of the parameters of the hyperplane then the body is an ellipsoid. This partially answers a question of V.I. Arnold: whether odd-dimensional ellipsoids are the only algebraically integrable bodies?

Let D be a bounded domain in Rn; with smooth boundary. Denote VD.!; t/; ! 2 Sn 1; t 2 R; the Radon transform of the characteristic function D of the domain D; i. e., the. n 1/- dimensional volume of the intersection D with the hyperplane fx 2 Rn W< !; x > D tg: If the domain D is an ellipsoid, then the function VD is algebraic and if, in addition, the dimension n is odd, then V.!; t/ is a polynomial with respect to t: Whether odd- dimensional ellipsoids are the only bounded smooth domains with such a property? The article is devoted to partial verification and discussion of this question.

It is proved that if a Paley-Wiener family of eigenfunctions of the Laplace operator in R-3 vanishes on a real-analytically ruled two-dimensional surface S subset of R-3 then S is a union of cones, each of which is contained in a translate of the zero set of a nonzero harmonic homogeneous polynomial. If S is an immersed C (1) manifold then S is a Coxeter system of planes. Full description of common nodal sets of Laplace spectra of convexly supported distributions is given. In equivalent terms, the result describes ruled injectivity sets for the spherical mean transform and confirms, for the case of ruled surfaces in R-3; a conjecture from [1].

It follows from the classical argument principle that if a holomorphic mapping f from the unit disc Δ to C or, more generally, to Cn, smooth in the closed disc, is homologically trivial on the unit circle (i.e. H1(γ)=0, γ=f(S1), which is equivalent to either γ being a point or ∂γ≠∅), then f=const, i.e. the image of the unit disc degenerates to a point. We establish a parametric version of this fact, for a variety of holomorphic mappings from Δ to Cn in place of a single mapping. We find conditions for a holomorphic mapping of the unit disc, depending on additional real parameters, under which homological triviality of the boundary image implies collapse of the dimension of the image of the interior. As an application, we obtain estimates of dimensions of complex tangent bundles of real submanifolds in Cn, in terms of zero moment conditions on families of closed curves covering the manifold. Applying this result to the graphs of functions, we obtain solution of several known problems about characterization of holomorphic CR functions in terms of moment conditions on families of curves.