关于内心单形的一个猜想

文献5]提出如下猜想:设n维Euclid空间En(n≥3)中n维单形∑A=conv{A0,A1,…,An}诸顶点Ai所对n-1维界面fi的内心为Ii(i=0,1,…,n),单形∑A与其内心单形∑I=conv{I0,I1,…,In}的有向体积分别为Vn(A)和Vn(I),则｜Vn(I)｜≤1nn｜Vn(A)｜等式成立当且仅当∑A为正则单形 本文利用垂心坐标与行列式计算证明了此猜想,同时放宽了猜想中所述不等式成立的充要条件

A Conjecture for the Incenter Simplex

In 5], Tang Li hua and Leng Gang song have guessed that the following inequality holds:Let∑A=conv｛A0,A1,…,An｝ be an n dimensional simplex in the n dimensional Euclidean space En(n≥3) with vertices A0,A1,…,An and of oriented volume Vn(A), and let fi=conv｛A0,A1,…,Ai-1,…,An｝ be its (n-1)dimensional face Let Bi be the incenter of an (n-1) dimensional face fi(i=0,1,2,…,n), and let ∑I=conv｛I0,I1,…,In｝ be an n dimensional simplex with vertices I0, I1,…,In and of oriented volume Vn(I). Then｜Vn(I)｜≤1nn｜Vn(A)｜,where equality holds if and only if ∑A is regular. In this paper, it has been proved that the above conjecture is valid by means of barycentic coordinates concept and the method of calculating determiant. Meanwhile, a necessary and sufficient condition that the above inequatity holds have been improved.