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

TL;DR

This paper investigates conditions under which reflection or rotational symmetries in finitely many subspaces imply full rotational symmetry in Euclidean space, providing new results for various dimensions and symmetry types.

Contribution

The paper extends existing results by characterizing when symmetries in subspaces lead to full rotational symmetry, especially for lower-dimensional subspaces, and describes the structure of rotation-invariant sets.

Findings

Reflection symmetry in certain subspaces implies full rotational symmetry in some cases.

Rotational symmetry about multiple axes constrains sets to be unions of spheres centered at the origin.

New results are obtained for subspaces of dimensions less than n-1, complementing previous work.

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.