PreprintCC BY 4.0arXiv
Algebraically Integrable Smooth Convex Bodies
Abstract
We prove that a convex body with C∞ boundary in an odd dimension n≥3 is algebraically integrable if and only if it is an ellipsoid. This answers Arnold's problem for smooth convex bodies. Together with Vassiliev's even-dimensional nonexistence theorem, it gives the classification in every dimension n≥2. More generally, in odd dimensions, a convex body with C∞ semialgebraic boundary is an ellipsoid whenever its section volume function satisfies a joint algebraic relation.
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BibTeX
@misc{shan2026algebraically,
title = {{Algebraically Integrable Smooth Convex Bodies}},
author = {Junjie Shan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19660},
eprint = {2610.19660},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19660},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Algebraically Integrable Smooth Convex Bodies
AU - Junjie Shan
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19660
UR - https://arxiv.org/abs/2610.19660
AB - We prove that a convex body with C^∞ boundary in an odd dimension n≥3 is algebraically integrable if and only if it is an ellipsoid. This answers Arnold's problem for smooth convex bodies. Together with Vassiliev's even-dimensional nonexistence theorem, it gives the classification in every dimension n≥2. More generally, in odd dimensions, a convex body with C^∞ semialgebraic boundary is an ellipsoid whenever its section volume function satisfies a joint algebraic relation.
ER -