Strongly convex metrics in cells
Strongly convex metrics in cells
复制标题
单元格中的强凸度量
DOI：
10.1090/s0002-9904-1968-11926-3
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发表时间：
1968
影响因子：
1.3
通讯作者：
D. Rolfsen
中科院分区：
文献类型：
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作者：
D. Rolfsen
Bing在[2]中提出了以下问题：“如果一个w维紧拓扑空间有一个强凸且没有分支的度量(定义如下)，它一定同胚于欧几里得w-胞格吗？”Lelek和Nitka[5]肯定地回答了这个问题，f或ng2；我们在下面给出一个证明，当n=3时，答案也是肯定的。虽然这个问题在高维空间中仍然是开放的，但当空间被假定为流形(=有边界的流形)和W5^4或5时，我们也给出了肯定的答案。实际上，在这个进一步的假设下，当n^3时，我们可以省略“无分支”的要求。如果X是一个空间，并且XTy，m，zx，则m称为x和y的中点，如果d(x，m)=d(m，y)=$d(x，y)=$d(x，y)，则m称为x和y的中点(关于X上的度量d)。如果每对点都有一个唯一的中点，并且没有分支(WR)，如果没有x的中点，并且y是x‘和y的中点，除非x=x，则度量是强凸的(SC)。这两个性质都是欧几里得空间和胞元上的常用度量所享有的，它们在笛卡尔乘积下保持如下意义：
The following question was raised by Bing in [2]: "If an w-dimensional compact topological space has a metric which is strongly convex and without ramifications (defined below), is it necessarily homeomorphic to the Euclidean w-cell?" Lelek and Nitka [5] answered this affirmatively f or n g 2 ; we outline below a proof that the answer is also yes when n = 3. Although the question remains open in higher dimensions, we also give an affirmative answer when the space is assumed to be a manifold ( = manifold with boundary) and W5^4 or 5. In fact with this further assumption we may omit the "without ramifications" requirement when n^3. If X is a space and xt y, mÇzX, then m is called a midpoint of x and y (with respect to a metric d on X) if d(x, m) = d(m, y) = $d(x, y). The metric is strongly convex (SC) if each pair of points has a unique midpoint and without ramifications (WR) if no midpoint of x and y is a midpoint of x' and y unless x = x. Both of these properties are enjoyed by the usual metric on Euclidean spaces and cells, and they are preserved under cartesian products in the following sense :