PreprintCC BY 4.0arXiv
On convex bodies with rotationally symmetric planar projections
Abstract
Let n≥ 3 and let K⊂ℝn be a convex body. For a two-dimensional linear subspace P⊂ℝn, let K|P be the orthogonal projection of K onto P. We prove that if for every two-dimensional subspace P, the planar convex body K|P has q-fold rotational symmetry up to translation, then for q≥4, this forces K to be an Euclidean ball. The case q=3 is exceptional: non-spherical examples exist.
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BibTeX
@misc{myroshnychenko2026convex,
title = {{On convex bodies with rotationally symmetric planar projections}},
author = {Sergii Myroshnychenko and Dmitry Ryabogin and Christos Saroglou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27192},
eprint = {2609.27192},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27192},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On convex bodies with rotationally symmetric planar projections
AU - Sergii Myroshnychenko
AU - Dmitry Ryabogin
AU - Christos Saroglou
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27192
UR - https://arxiv.org/abs/2609.27192
AB - Let n≥ 3 and let K⊂ℝ^n be a convex body. For a two-dimensional linear subspace P⊂ℝ^n, let K|P be the orthogonal projection of K onto P. We prove that if for every two-dimensional subspace P, the planar convex body K|P has q-fold rotational symmetry up to translation, then for q≥4, this forces K to be an Euclidean ball. The case q=3 is exceptional: non-spherical examples exist.
ER -