Abstract: 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.

Key words: Euclidean space, Synclastic simplexes, Isometric embedding, Generalized metric addition, Geometry inequality