Matheron’s covariogram uniqueness problem in three dimensions
For a convex body $K \subset \mathbb{R}^3$, that is, a compact convex set with nonempty interior, the covariogram is $$g_K(x) = \operatorname{Vol}_3\bigl(K \cap (K + x)\bigr), \qquad x \in \mathbb{R}^3.$$ It is the autocorrelation of $\mathbf{1}_K$, so its Fourier transform is $|\widehat{\mathbf{1}_K}|^2$, and it does not change when $K$ is translated or reflected in a point. In the plane the covariogram determines a convex body up to these motions: Nagel proved this for convex polygons, and Bianchi and Averkov and Bianchi for all planar convex bodies. In every dimension $n \ge 4$ Bianchi found pairs of convex bodies, which may be polytopes, with the same covariogram that are not translates or reflections of each other. In $\mathbb{R}^3$, a theorem of Goodey, Schneider and Weil on polytopes implies that most convex bodies, in the sense of Baire category, are determined among all convex bodies; Bianchi proved that convex polytopes are, and that two convex bodies of class $C^9_+$ with the same covariogram coincide up to translation and reflection.
Problem. Let $K, L \subset \mathbb{R}^3$ be convex bodies with $g_K(x) = g_L(x)$ for every $x \in \mathbb{R}^3$. Must $L = a + K$ or $L = a - K$ for some $a \in \mathbb{R}^3$? No smoothness, curvature or polyhedral assumption is made on $K$ or $L$.
Source: Matheron (1986), p. 20, who asked it in every dimension and conjectured a positive answer in the plane; see Bianchi (2023), §1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MG — Metric Geometry
- Secondary
- CA
- Keywords
- AIM Open Applied Problems, convex bodies, covariograms, phase retrieval
References
@techreport{matheron1986,
author = {Matheron, Georges},
title = {Le covariogramme g{\'e}om{\'e}trique des compacts convexes des {$\mathbb{R}^2$}},
institution = {Centre de G{\'e}ostatistique, {\'E}cole Nationale Sup{\'e}rieure des Mines de Paris},
number = {N-2/86/G},
year = {1986},
}
@incollection{bianchi2023,
author = {Bianchi, Gabriele},
title = {The covariogram problem},
booktitle = {Harmonic Analysis and Convexity},
editor = {Koldobsky, Alexander and Volberg, Alexander},
publisher = {De Gruyter},
address = {Berlin},
year = {2023},
pages = {37--82},
doi = {10.1515/9783110775389-002},
}
@article{nagel1993,
author = {Nagel, Werner},
title = {Orientation-dependent chord length distributions characterize convex polygons},
journal = {J. Appl. Probab.},
volume = {30},
number = {3},
year = {1993},
pages = {730--736},
doi = {10.1017/S0021900200044442},
}
@article{bianchi2005,
author = {Bianchi, Gabriele},
title = {Matheron's conjecture for the covariogram problem},
journal = {J. London Math. Soc. (2)},
volume = {71},
number = {1},
year = {2005},
pages = {203--220},
doi = {10.1112/S0024610704006039},
}
@article{averkov2009,
author = {Averkov, Gennadiy and Bianchi, Gabriele},
title = {Confirmation of {M}atheron's conjecture on the covariogram of a planar convex body},
journal = {J. Eur. Math. Soc. (JEMS)},
volume = {11},
number = {6},
year = {2009},
pages = {1187--1202},
doi = {10.4171/JEMS/179},
}
@article{goodey1997,
author = {Goodey, Paul and Schneider, Rolf and Weil, Wolfgang},
title = {On the determination of convex bodies by projection functions},
journal = {Bull. London Math. Soc.},
volume = {29},
number = {1},
year = {1997},
pages = {82--88},
doi = {10.1112/S0024609396001968},
}
@article{bianchi2009,
author = {Bianchi, Gabriele},
title = {The covariogram determines three-dimensional convex polytopes},
journal = {Adv. Math.},
volume = {220},
number = {6},
year = {2009},
pages = {1771--1808},
doi = {10.1016/j.aim.2008.11.011},
eprint = {0805.1605},
archivePrefix = {arXiv},
}
@article{bianchi2016,
author = {Bianchi, Gabriele},
title = {The covariogram and {F}ourier--{L}aplace transform in {$\mathbb{C}^n$}},
journal = {Proc. Lond. Math. Soc. (3)},
volume = {113},
number = {1},
year = {2016},
pages = {1--23},
doi = {10.1112/plms/pdw020},
}
@misc{aim-three-dimensional-covariogram-uniqueness,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Matheron’s covariogram uniqueness problem in three dimensions}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/261-three-dimensional-covariogram-uniqueness.md},
note = {Entry 261 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-00567, "Matheron’s covariogram uniqueness problem in three dimensions", clopen.solutions.