n维Oppenheim不等式及其应用

n-DIMENSIONAL OPPENHEIM'S INEQUALITY AND ITS APPLICATIONS

设ａ＿ｉ（ｉ＝１，２，３）为三角形ΔＡ＿１Ａ＿２Ａ＿３的边长，Ｓ为ΔＡ＿１Ａ＿２Ａ＿３的面积，λ＿ｊ（ｊ＝１，２，３）为任一组正数．作者将Ｏｐｌｓｏｎｂｅｉｍ三角形不等式推广到ｎ维欧氏空间Ｅ￣ｎ中的ｎ维单形，从而获得了ｎ维单形的Ｏｐｐｅｎｂｅｉｍ不等式这里Ｖ是ｎ维单形Ａ＿１Ａ＿２…Ａ＿（ｎ＋１）的体积，Ｖ＿ｉ为顶点Ａ＿ｉ所对之侧面的面积，λ＿ｉ为任意一组正数．

SppoSe a_i(I=1,2,3 )are lengths of three edges of triangle ΔA_1A_2A_3 with area S and λ_i(i=1,2,3 )are three positive numbers.The triangle inequity in Euclid space of dimension 3is generalized to simplex in Euclid space of dimension n ,and further come to the Oppenheim inequityof simplex of dimension n :Where V is the volume of simplex A_1A_2…A_(n+1) of dimension n,and V_i is the area of lateral surface oppo-site to vertice A_i andλ_i are arbitrary positive numbers for i=1,2,…,n+1.