同向单形到欧氏空间的等距嵌入及其应用

该文利用矩阵的方法, 获得了两个同向的 n 维单形同时等距嵌入 Eⁿ 维欧氏空间的一个充分必要条件是: 对于预给(n+1)²个距离,满足一组具有行列式形式的不等式组det(△_k)<0, 由此可以得到两组等数量的有限点集合到 Eⁿ 维欧氏空间中等长嵌入的一个充分必要条件. 然后利用杨路和张景中引进的代数方法, 应用广义等距嵌入定理, 提出了关于两组两个完全同向的 n 维单形“广义度量加”的概念, 并且证明了涉及“广义度量加”的一个几何不等式, 它推广了杨路和张景中关于Alexander猜想的结果. 同时我们将杨路和张景中关于Neuberg-Pedoe不等式的高维推广形式推广到两组两个完全同向的 n 维单形中, 获得了涉及四个单形的一类几何不等式, 它们蕴含近期诸多文献的主要结果.

The Isometric Embedding of Synclastic Simplexes to Euclidean Space and Applications

In this paper, by the matrix method, a necessary and sufficient condition for two n-dimensional synclastic simplexes to be able embedded in the n-dimensional Euclidean space is obtained, as well as for two group finite point sets. Then, by using the algebraic method which was introduced by Yang L. and Zhang J. Z., and the generalized isometric embedding theory, the generalized concept of metric addition is brought forward, a geometric inequality about the generalized metric addition is given, and the main result of Yang L. and Zhang J. Z. on Alexander conjecture is generalized, the higher dimensional Neuberg-Pedoe inequality generalized by Yang L. and Zhang J. Z. is generalized also. By these inequalities, a class of geometric inequalities about four simplexes is obtained, These imply some recent results in literature.