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arXiv · 2202.04004

Full rotational symmetry from reflections or rotational symmetries in finitely many subspaces

Gabriele Bianchi·Richard J. Gardner·Paolo Gronchi
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Abstract

Two related questions are discussed. The first is when reflection symmetry in a finite set of $i$-dimensional subspaces, $i\in \{1,\dots,n-1\}$, implies full rotational symmetry, i.e., the closure of the group generated by the reflections equals $O(n)$. For $i=n-1$, this has essentially been solved by Burchard, Chambers, and Dranovski, but new results are obtained for $i\in \{1,\dots,n-2\}$. The second question, to which an essentially complete answer is given, is when (full) rotational symmetry with respect to a finite set of $i$-dimensional subspaces, $i\in \{1,\dots,n-2\}$, implies full rotational symmetry, i.e., the closure of the group generated by all the rotations about each of the subspaces equals $SO(n)$. The latter result also shows that a closed set in $\mathbb{R}^n$ that is invariant under rotations about more than one axis must be a union of spheres with their centers at the origin.

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Gabriele Bianchi, Richard J. Gardner, Paolo Gronchi. 2022-02-08. Full rotational symmetry from reflections or rotational symmetries in finitely many subspaces. https://doi.org/10.1512/iumj.2022.71.9818

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