Let A={A_1, A_2,…, A_(n+1)} be a simplex in E^n which its center O of circumscribed sphere is in inside of A. If R and R_i are radiuses of A_i respectively (A_i={A_1, A_2,…, A_(i-1), O, A_(i+1),…,A_(n+1)} ,i=1,2,…...Let A={A_1, A_2,…, A_(n+1)} be a simplex in E^n which its center O of circumscribed sphere is in inside of A. If R and R_i are radiuses of A_i respectively (A_i={A_1, A_2,…, A_(i-1), O, A_(i+1),…,A_(n+1)} ,i=1,2,…,n+1),then we have The equality holds if and only if A is a regular simplex.展开更多
In this paper, the concept of a finite mass-points system ΣN(H(A))(N > n) being in a sphere in an n-dimensional hyperbolic space H n and a finite mass-points system ΣN(S(A))(N > n) being in a hyperplane in an ...In this paper, the concept of a finite mass-points system ΣN(H(A))(N > n) being in a sphere in an n-dimensional hyperbolic space H n and a finite mass-points system ΣN(S(A))(N > n) being in a hyperplane in an n-dimensional spherical space S n is introduced, then, the rank of the Cayley-Menger matrix-ΛN(H) (or a-ΛN(S)) of the finite mass-points system ΣN(S(A)) (or ΣN(S(A))) in an n-dimensional hyperbolic space H n (or spherical space S n) is no more than n + 2 when ΣN(H(A))(N > n) (or ΣN(S(A))(N > n)) are in a sphere (or hyperplane). On the one hand, the Yang-Zhang’s inequalities, the Neuberg-Pedoe’s inequalities and the inequality of the metric addition in an n-dimensional hyperbolic space H n and in an n-dimensional spherical space S n are established by the method of characteristic roots. These are basic inequalities in hyperbolic geometry and spherical geometry. On the other hand, some relative problems and conjectures are brought.展开更多
文摘Let A={A_1, A_2,…, A_(n+1)} be a simplex in E^n which its center O of circumscribed sphere is in inside of A. If R and R_i are radiuses of A_i respectively (A_i={A_1, A_2,…, A_(i-1), O, A_(i+1),…,A_(n+1)} ,i=1,2,…,n+1),then we have The equality holds if and only if A is a regular simplex.
基金This work was partially supported by the National Key Basic Research Project of China(Grant No.2004CB318003).
文摘In this paper, the concept of a finite mass-points system ΣN(H(A))(N > n) being in a sphere in an n-dimensional hyperbolic space H n and a finite mass-points system ΣN(S(A))(N > n) being in a hyperplane in an n-dimensional spherical space S n is introduced, then, the rank of the Cayley-Menger matrix-ΛN(H) (or a-ΛN(S)) of the finite mass-points system ΣN(S(A)) (or ΣN(S(A))) in an n-dimensional hyperbolic space H n (or spherical space S n) is no more than n + 2 when ΣN(H(A))(N > n) (or ΣN(S(A))(N > n)) are in a sphere (or hyperplane). On the one hand, the Yang-Zhang’s inequalities, the Neuberg-Pedoe’s inequalities and the inequality of the metric addition in an n-dimensional hyperbolic space H n and in an n-dimensional spherical space S n are established by the method of characteristic roots. These are basic inequalities in hyperbolic geometry and spherical geometry. On the other hand, some relative problems and conjectures are brought.
