论文标题
在有限的许多子空间中，反射或旋转对称性的完全旋转对称性
Full rotational symmetry from reflections or rotational symmetries in finitely many subspaces
论文作者
论文摘要
讨论了两个相关问题。首先是在有限的$ i $二维子空间中的反射对称性，$ i \ in \ {1，\ dots，n-1 \} $，意味着完全旋转对称性，即，由反射量等于$ o（n）$产生的组的关闭。对于$ i = n-1 $，本质上是由Burchard，Chambers和Dranovski解决的，但是在\ {1，\ dots，n-2 \} $中获得了新的结果。 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 $SO(n)$.后一个结果还表明，在旋转大约多个轴的旋转下不变的$ \ Mathbb {r}^n $中的封闭设置必须是与其中心处的球体结合。
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.
