ZHOU Yong-guo

(Mingda middle school,Changsha,Hunan,410000,China)

Abstract:In this paper,by using the theory and method of distance geometry,we study the geometric inequality of a n-dimensional simplex in the spherical space and establish two geometric inequalities involving the edge-length and volume of one simplex and the volume,height and(n-1)-dimensional volume of the side of another simplex in the n-dimensional spherical space.They are the extensions of the results[10]in the n-dimensional Euclidean geometry to the n-dimensional spherical space.

Key words:spherical space;simplex;volume;edge-length;height;inequality

§1.Introduction and Main Results

Research achieveents on geometric inequalities in high-dimensional Euclidean space have been fruitful since 1980s in the 20th Century([1]).In particular,the work of Chinese mathematicians Lu Yang and Jingzhong Zhang on establishing several profound and elegant inequalities has been highly recognized nationwide by the international society([2-6]).In the last two decades,researchers on distance geometry attempted to extend these inequalities in Euclidean space.However,the development of high-dimensional geometric inequalities in non-Euclidean space was quite slow.In 2006,Professor Dinghua Yang([7])established several fundamental geometric inequalities in high-dimensional non-Euclidean space.Later,Professor Shiguo Yang([8])obtained several geometric inequalities on the volume,circumradius,inradius and edgelength of an n-dimensional simplex in the spherical space.More recently,Shiguo Yang obtained the Zhang-Yang type inequality([9])in n-dimensional spherical space.In this paper,we study the geometric inequality problems involving two n-dimensional simplexes in the spherical space,and establish two geometric inequalities on the edge-length and volume of one simplex and the volume,height and(n-1)-dimensional volume of the side of another simplex in n-dimensional spherical space.They are the extensions of the existing results in n-dimensional Euclidean geometry([10])to that in n-dimensional spherical space.

Let Ψk=Ak1,Ak2,···,Ak(n+1)(k=1,2)denote two n-dimensional simplexes in ndimensional Euclidean space,with edge-length|AkiAkj|=akij(k=1,2;0≤i

Suppose the set of all points on the sphere with origin o and radius ρ in(n+1)-dimensional Euclidean space is

For arbitrary two points x(x1,x2,···,xn+1)and y(y1,y2,···,yn+1)in Sn,we defines the distance to be the smallest non-negative real number satisfying

Let Ω ={A0,A1,···,An}and Ω′={A′0,A′2,···,A′n}be two n-dimensional spherical space Sn(K)with edge-length=aijand=a′ij(0≤ i

In this work we study the inequality relationship between the geometric quantities edgelength,volume,(n-1)-dimensional volume and height of the side for two simplexes in spherical space.We establish the following two geometric inequalities.

Theorem 1 For two n-dimensional simplexes Ω, Ω′in n-dimensional spherical space Sn(K),we have

The equality in(1.4)holds when Ω,Ω′are both regular simplex.

Theorem 2 For two n-dimensional simplexes Ω, Ω′in n-dimensional spherical space Sn(K),we have

The equality in(1.5)holds when both Ω,Ω′are regular simplexes.

§2.Proof of Lemma and Theorem

In order to prove the two theorems above,we introduce the following lemmas:

Lemma 1[9]For n-dimensional simplex Ω in Sn(K)and a real number mi＞ 0(i=0,1,···,n),we have

The equality in(2.1)holds when m0=m1=L=mnand Ω is a regular simplex.

Note Lemma 1 is Corollary 5 in reference[9],the original layout was incorrect.

Lemma 2[8]For n-dimensional simplex Ω in Sn(K),we have

The equality in(2.2)holds when Ω is a regular simplex.

Lemma 3[9]For n-dimensional simplex Ω in Sn(K),we have

The equality in(2.3)holds when Ω is a regular simplex.

Proof of Theorem 1 Let mi=sini=1,2,···,n)in inequality(2.1),we have

Plugging inequality(2.2)into the equation above yields inequality(1.4).In view of the equality conditions in(2.1)and(2.2),the equality condition in(1.4)is as described in Theorem 1.

Proof of Theorem 2 In(2.1),letwe have

It follows from(2.3)that

Pugging it into the inequality above yields inequality(1.5).In view of the equality conditions in(2.1)and(2.3),the equality condition in(1.5)is as described in Theorem 2.

In Theorem 1 and 2,letting Ω′= Ω,we obtain

Corollary 1 For n-dimensional simplex Ω in n-dimensional spherical space Sn(K),we have

The equality in(2.5)holds when Ω is a regular simplex.

Corollary 2 For n-dimensional simplex Ω in n-dimensional spherical space Sn(K),we have

The equality in(2.6)holds when Ω is a regular simplex.

AcknowledgementThe author wishes to express sincere gratitude to Professor Gangsong Leng for his generous advice and Dr.Liangying Ma for his assistance and help.