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, we use a geometric identity in the n-dimensional Euclidean space En and give the further improveme nt of Klamkin inequality in the space En.
The problem on the geometrc inequalities involving an n-dimensional simplex and its inscribed simplex is studied. An inequality is established, which reveals that the difference between the squared circumradius of the...The problem on the geometrc inequalities involving an n-dimensional simplex and its inscribed simplex is studied. An inequality is established, which reveals that the difference between the squared circumradius of the n-dimensional simplex and the squared distance between its circumcenter and barycenter times the squared circumradius of its inscribed simplex is not less than the 2(n-1)th power of n times its squared inradius, and is equal to when the simplex is regular and its inscribed siplex is a tangent point one. Deduction from this inequality reaches a generalization of n-dimensional Euler inequality indicating that the circumradius of the simplex is not less than the n-fold inradius. Another inequality is derived to present the relationship between the circumradius of the n-dimensional simplex and the circumradius and inradius of its pedal simplex.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.
文摘In this paper, we use a geometric identity in the n-dimensional Euclidean space En and give the further improveme nt of Klamkin inequality in the space En.
文摘The problem on the geometrc inequalities involving an n-dimensional simplex and its inscribed simplex is studied. An inequality is established, which reveals that the difference between the squared circumradius of the n-dimensional simplex and the squared distance between its circumcenter and barycenter times the squared circumradius of its inscribed simplex is not less than the 2(n-1)th power of n times its squared inradius, and is equal to when the simplex is regular and its inscribed siplex is a tangent point one. Deduction from this inequality reaches a generalization of n-dimensional Euler inequality indicating that the circumradius of the simplex is not less than the n-fold inradius. Another inequality is derived to present the relationship between the circumradius of the n-dimensional simplex and the circumradius and inradius of its pedal simplex.
