Broken-ray tomography with two smooth convex obstacles
Let $\Omega \subset \mathbb{R}^2$ be a bounded smooth strictly convex domain, and let $K_1, K_2 \Subset \Omega$ be disjoint compact convex sets with smooth boundaries, at least one of them strictly convex. Put $M = \overline{\Omega} \setminus (K_1^\circ \cup K_2^\circ)$. A measured broken ray is a finite piecewise straight unit-speed path in $M$ that starts and ends on $\partial\Omega$, meets $\partial\Omega$ nowhere in between, and reflects specularly on $\partial K_1 \cup \partial K_2$: the outgoing velocity is the incoming one minus twice its normal component. Straight segments without reflection are included; rays touching an obstacle tangentially are discarded. For $f \in C^\infty(M)$ and a measured broken ray $\gamma$ put $Bf(\gamma) = \int_\gamma f\,ds$. Eskin proved injectivity of the broken ray transform with several obstacles in the plane, under geometric assumptions that exclude smooth obstacles. Ilmavirta and Salo proved that on a nonpositively curved surface with one strictly convex obstacle a function is determined by its integrals over broken rays reflecting on the obstacle. If the two obstacles are only assumed convex, $B$ can fail to be injective (Ilmavirta).
Problem. For every such $\Omega$, $K_1$ and $K_2$, does $Bf(\gamma) = 0$ for every measured broken ray $\gamma$ imply $f = 0$ on $M$?
Source: Ilmavirta (2014), doctoral dissertation, introduction, pp. 20–21; see Jathar and Railo (2025), §2, open problem (i).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DG
- Keywords
- AIM Open Applied Problems, broken ray transforms, integral geometry, reflecting obstacles
References
@phdthesis{ilmavirta2014,
author = {Ilmavirta, Joonas},
title = {On the broken ray transform},
school = {University of Jyv{\"a}skyl{\"a}},
year = {2014},
note = {Report 140, Department of Mathematics and Statistics; ISBN 978-951-39-5743-8},
url = {https://users.jyu.fi/~jojapeil/thesis/brt_290814.pdf},
eprint = {1409.7500},
archivePrefix = {arXiv},
}
@incollection{jathar2025,
author = {Jathar, Shubham R. and Railo, Jesse},
title = {Survey on broken ray transforms},
booktitle = {Inverse Problems: Modelling and Simulation},
series = {Trends in Mathematics},
volume = {11},
publisher = {Birkh{\"a}user},
address = {Cham},
year = {2025},
pages = {51--56},
doi = {10.1007/978-3-031-87213-6_7},
eprint = {2410.04908},
archivePrefix = {arXiv},
}
@article{eskin2004,
author = {Eskin, G.},
title = {Inverse boundary value problems in domains with several obstacles},
journal = {Inverse Problems},
volume = {20},
number = {5},
year = {2004},
pages = {1497--1516},
doi = {10.1088/0266-5611/20/5/011},
}
@article{ilmavirta2016,
author = {Ilmavirta, Joonas and Salo, Mikko},
title = {Broken ray transform on a {R}iemann surface with a convex obstacle},
journal = {Comm. Anal. Geom.},
volume = {24},
number = {2},
year = {2016},
pages = {379--408},
doi = {10.4310/CAG.2016.v24.n2.a6},
eprint = {1403.5131},
archivePrefix = {arXiv},
}
@article{ilmavirta2015,
author = {Ilmavirta, Joonas},
title = {A reflection approach to the broken ray transform},
journal = {Math. Scand.},
volume = {117},
number = {2},
year = {2015},
pages = {231--257},
doi = {10.7146/math.scand.a-22869},
eprint = {1306.0341},
archivePrefix = {arXiv},
}
@misc{aim-two-obstacle-broken-rays,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Broken-ray tomography with two smooth convex obstacles}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/060-two-obstacle-broken-rays.md},
note = {Entry 060 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-00568, "Broken-ray tomography with two smooth convex obstacles", clopen.solutions.