一个几何不等式的注记

A NOTE ON A GEOMETRIC INEQUALITY

设A是n维欧氏空间E^n中的一个非退化单形,它的切点单形为B,又以A的顶点和它的内切球心的连线与A的n+1个侧面的交点为顶点的单形设为D。若单形A,B,D的体积分别记为V(A),V(B),V(D),本文证明了V(B)≤V(D)≤(1/n^n)V(A)。

Let A. (i=1, 2, …. n+1) be the vertex of a simplex ? in n-dimensionalEuclidean space E^n, B_i (i=1, 2, …, n+1) be the tangent points at which theinscribed sphere of ? is tangent to the side faces of ?. Then the simplex ?with the tangent points as vertexes is called the tangent points simplex. Thestraight line A_iI intersects the surface A_1A_2…A_(i-1)A_(i+1)…A_(n+1) of ? at the pointD_i Where I is the inner center of ?. Let ? be the simplex with the D_i (i=1,2,…, n+1) as its vertexes and V(?), V(?), V(?) be the volume of ?, ?,?, the following inequality is proved:V(?)≤V (?)≤(1/n^n)V(?).