论文标题
BANACH的等距子空间问题四大
Banach's isometric subspace problem in dimension four
论文作者
论文摘要
我们证明，如果凸面$ b \ subset \ mathbb r^4 $带有3维线性子空间的所有交叉点是线性等效的，那么$ b $是一个集中的椭圆形。这给了Banach从1932年开始对案件的肯定答案：$ n $ d $维线性子空间都是等值的，对于固定的$ 2 \ le n <\ dim v $，一定是euclidean？ 尺寸$ n = 3 $和$ \ dim v = 4 $是问题未解决的第一个情况。由于$ 3 $ -sphere是可行的，因此在这种情况下，已知的全球拓扑方法无济于事。我们的证明采用了差异几何方法。
We prove that if all intersections of a convex body $B\subset\mathbb R^4$ with 3-dimensional linear subspaces are linearly equivalent then $B$ is a centered ellipsoid. This gives an affirmative answer to the case $n=3$ of the following question by Banach from 1932: Is a normed vector space $V$ whose $n$-dimensional linear subspaces are all isometric, for a fixed $2 \le n< \dim V$, necessarily Euclidean? The dimensions $n=3$ and $\dim V=4$ is the first case where the question was unresolved. Since the $3$-sphere is parallelizable, known global topological methods do not help in this case. Our proof employs a differential geometric approach.
