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

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.

16 pages, 2 figures. The core of this paper was Section 3 of arXiv:1908.03259v2 and will be published separately following the suggestion of a referee. Some proofs are shorter and clearer (Lemma 3.1 and Theorem 3.2), others are completely different and much shorter (the symmetry extension lemma, now Lemma 3.7). In version 2 some typos have been corrected