关于单形的一类不等式

On a Kind of Inequalities of Simplex

本文证明了n维单形的一类不等式。设B_1是n维单形A_1A_2…A_(n+1)的任-n-1维平面X内的任意一点,过B_1作不在该面上的各棱的平行线交其余各面于B_2,B_3,…B_(n+1)则:|V_(B_1B_2…B_(n+1)|≤1/n^n|V_(A_1A_2…A_(n+1)|,式中等号当且仅当B_1是面X的重心时成立。

In this paper,We obtain several inequalities for an n -dimensional simplex. Let B1. be a point in an N -1- dimensional face X of the n -dimensional simplex A1A2…An-1. Draw lines through BI parallel to every side which is not in the face X . Let ponits of intersection of the parallel lines and an other face be B2,B3…Bn-1 respectiuely,Then |VB1B2…Bn+1|≤1\nn|VA1A2…An+1| In whicn the equality sign holds if and only if BI is the barycenter of the face X.