三维欧氏空间中的单位球, 如果去掉中心, 则可以分裂为四个不相交的子集. 通过旋转可以重新组合成两个这样的去心单位球.

Posted by haifeng on 2020-09-18 17:59:11 last update 2020-09-18 21:23:22 | Edit | Answers (0)

设 $B$ 是三维欧氏空间 $\mathbb{R}^3$ 中的去心单位球, 即

\[B=\{(x,y,z)\in\mathbb{R}^3\mid 0 < x^2+y^2+z^2\leqslant 1\}\]

则 $B$ 可以分裂为四个不相交的子集, 它们通过旋转可以重新组合以形成两个 $B$ 的拷贝.




Q. 如何构造这四个野集? 通过怎样的旋转?

参考自 Dmitri Burago, Yuri Burago, Sergei Ivanov 著《A Course in Metric Geometry》之 1.7 节.
1.7 Hausdorff Measure and Dimension

Let $B$ denote a unit ball in $\mathbb{R}^3$ with its center removed. Then $B$ can be split into four disjoint subsets, which can be rearranged (by means of rotations) so as to form two copies of $B$.

These four subsets are in fact extremely wild sets.