Let S=A_oA_1…A. be a simplex in n-dimensional Euclidean Space E^n with centroid G. The straight line A,G intersects the surface A_0A_1…Ai_l A_i+l…A. of S at the point G, and the circumsphere F of Sat the point Ai(i...Let S=A_oA_1…A. be a simplex in n-dimensional Euclidean Space E^n with centroid G. The straight line A,G intersects the surface A_0A_1…Ai_l A_i+l…A. of S at the point G, and the circumsphere F of Sat the point Ai(i=0,1,…,n). Let the edge-lengths of S be a_ij=A_iA_j(i,j=0,1,…,n, i≠j) and the me dians be m,=A_iG_i(i=O,1,…,n). Following theorem will be proved. Theorem. For the simplex in E^n, wo have sum from i=0 to x A_iA_i≥2n/n+1t=1 sum from i=0 to x m_i; (1) sum from i=0 to x A_i A_i≥2^(2/1)~n(N+1) sum from 0≤i<f≤x a_if; (2) sum from i=v to x (A_i A_i)~2≥4n+1 sum from 0≤i<f≤x ai; (3) The equalities in (1) and (3) hold if and only if the centroid G and the center O of the circumsphere of S are concurrent. The equality in (2) holds if and only if S is regular simplex.展开更多
文摘Let S=A_oA_1…A. be a simplex in n-dimensional Euclidean Space E^n with centroid G. The straight line A,G intersects the surface A_0A_1…Ai_l A_i+l…A. of S at the point G, and the circumsphere F of Sat the point Ai(i=0,1,…,n). Let the edge-lengths of S be a_ij=A_iA_j(i,j=0,1,…,n, i≠j) and the me dians be m,=A_iG_i(i=O,1,…,n). Following theorem will be proved. Theorem. For the simplex in E^n, wo have sum from i=0 to x A_iA_i≥2n/n+1t=1 sum from i=0 to x m_i; (1) sum from i=0 to x A_i A_i≥2^(2/1)~n(N+1) sum from 0≤i<f≤x a_if; (2) sum from i=v to x (A_i A_i)~2≥4n+1 sum from 0≤i<f≤x ai; (3) The equalities in (1) and (3) hold if and only if the centroid G and the center O of the circumsphere of S are concurrent. The equality in (2) holds if and only if S is regular simplex.
