Classify noninjectivity sets of spherical mean tomography
For $d \ge 3$ and $f \in C_c(\mathbb{R}^d)$ let $$Rf(x, r) = \int_{S^{d-1}} f(x + r\omega)\,d\omega, \qquad x \in \mathbb{R}^d,\ r > 0,$$ be the spherical mean transform. A set $S \subset \mathbb{R}^d$ is a noninjectivity set if some nonzero $f \in C_c(\mathbb{R}^d)$ satisfies $Rf(x, r) = 0$ for all $x \in S$ and $r > 0$. The zero set of a nonzero homogeneous harmonic polynomial, and any of its translates, is a noninjectivity set, and so is every real algebraic set of dimension at most $d - 2$. In the plane, Agranovsky and Quinto proved that the noninjectivity sets are exactly the subsets of the image, under a rigid motion, of a Coxeter system of lines (finitely many lines through a point at equal angles) together with a finite set. Agranovsky, Volchkov and Zalcman proved that a cone with vertex at the origin is a noninjectivity set if and only if it is contained in the zero set of a nonzero homogeneous harmonic polynomial. Agranovsky proved that a real-analytically ruled surface in $\mathbb{R}^3$ that is a noninjectivity set is a union of cones, each contained in a translate of the zero set of a nonzero homogeneous harmonic polynomial.
Problem. Is every noninjectivity set contained in a set of the form $$(a + \{x : h(x) = 0\}) \cup V,$$ where $a \in \mathbb{R}^d$, $h$ is a nonzero homogeneous harmonic polynomial of positive degree and $V$ is a real algebraic set of dimension at most $d - 2$?
Source: Agranovsky and Quinto (1996); see Agranovsky (2017), Conjecture 4.2 in the arXiv version, and Kuchment and Kunyansky (2008), Conjecture 4. AIM — Open Applied Problems states its necessity direction for d ≥ 3.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- CA
- Keywords
- AIM Open Applied Problems, harmonic polynomials, injectivity sets, spherical means
References
@article{agranovsky1996,
author = {Agranovsky, Mark L. and Quinto, Eric Todd},
title = {Injectivity sets for the {R}adon transform over circles and complete systems of radial functions},
journal = {J. Funct. Anal.},
volume = {139},
number = {2},
year = {1996},
pages = {383--414},
doi = {10.1006/jfan.1996.0090},
}
@article{agranovsky2017,
author = {Agranovsky, Mark L.},
title = {Ruled nodal surfaces of {L}aplace eigenfunctions and injectivity sets for the spherical mean {R}adon transform in $\mathbb{R}^3$},
journal = {J. Spectr. Theory},
volume = {7},
number = {4},
year = {2017},
pages = {1039--1099},
doi = {10.4171/JST/185},
eprint = {1504.01250},
archivePrefix = {arXiv},
}
@article{kuchment2008,
author = {Kuchment, Peter and Kunyansky, Leonid},
title = {Mathematics of thermoacoustic tomography},
journal = {European J. Appl. Math.},
volume = {19},
number = {2},
year = {2008},
pages = {191--224},
doi = {10.1017/S0956792508007353},
eprint = {0704.0286},
archivePrefix = {arXiv},
}
@article{agranovsky1999,
author = {Agranovsky, M. L. and Volchkov, V. V. and Zalcman, L. A.},
title = {Conical uniqueness sets for the spherical {R}adon transform},
journal = {Bull. London Math. Soc.},
volume = {31},
number = {2},
year = {1999},
pages = {231--236},
doi = {10.1112/S0024609398005396},
}
@misc{aim-spherical-mean-noninjectivity-sets,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Classify noninjectivity sets of spherical mean tomography}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/058-spherical-mean-noninjectivity-sets.md},
note = {Entry 058 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00562, "Classify noninjectivity sets of spherical mean tomography", clopen.solutions.