摘要
蒋星耀、刘根洪分别在[1]、[2]中建立了欧式空间E^m中关于单形顶点角正弦的不等式.本文首先利用代数方法证明一个涉及单形顶点角与二面角的不等式(定理1),然后再给出双曲型空间H^n中单形二面角的一个不等式(定理2).作为定理1的特例,可导出[1],[2]的某些主要结果.利用定理1,我们还可以将Klamkin提出的一个不等式推广到E^m.
Theorem 1 We show that let S be a simplex in En, its vertex angles being αi and interior dihedral angles formed by arbitrary two side faces fi,fj of S being Qij, if mi are positive numbers (i,j= 1, 2, …, n + 1, i ≠ j), thenTheorem 2 Let Ω be a simplex in Hn, its interior dihedral angles formed by arbitrary two side faces Fi, Fj of Ω being φij, if pi- be positive numbers (i, j = 1,2,…., n + 1, i ≠ j), then
