【摘要】 在欧氏空间Eⁿ中,借助于代数不等式和数学归纳法,将n维欧氏空间Eⁿ中单形∑（E,n）二面角的一类参数不等式做了进一步的推广,建立了涉及2^-m（m∈N）倍二面角θ_ij^E的一类无约束型参数的不等式.作为这些不等式的应用,将一些三角形中常用三角不等式推广到n（>2）维单形的情形,与此同对还提出了一个尚待解决的猜想.在双曲空间Hⁿ中,借助于n维双曲空间Hⁿ的射影模型,提出了单形∑（H,n）所有侧面共超球的概念,在这个条件下,证明了二次型I（x）=∑_i=0ⁿ∑_j=0ⁿcosθ_ij^Hx_ix_j在约束条件x₀+x₁+…+x_n=0下是正定的,然后应用矩阵的次特征值理论,给出了单形∑（H,n）内切超球半径为r（H）的一个极值表示,导出了所有超平面共超球的单形∑（H,n）二面角θ_ij^H的一类具有约束型的参数不等式.更多还原

【Abstract】 In this paper,making use of algebraic inequalities and mathematical induction, a class of parametric inequalities of the dihedral angles of simplex∑（E,n）in n-dimensional Eu- clidean space Eⁿ are gerenalized,a class of parametric inequalities without constraint condition about 2-mθ_ij^E（m∈N）are established.As their applications,some triangle inequalities are gerenalized to higher dimensional Euclidean space Eⁿ,and a unsolved problem is introduced. The concept of all lateral face of simplex∑（H,n）in a hypersphere is introduced by the model of n-dimensional hyperbolic space Hⁿ,then the quadratic form I（x）=∑_i=0ⁿ∑_j=0ⁿcosθ_ij^Hx_ix_j with constraint condition x₀+x₁+…+x_n=0 is positive definite form when all lateral face of simplex∑（H,n）is in a hypersphere,a extremal representation of the radius of inscribed sphere of∑（H,n）is given,a class of parametric inequalities with constraint condition are established.更多还原