# Inequalities for an n-simplex in spherical space ${S}_{n}\left(1\right)$

## Abstract

For an n-dimensional simplex ${\mathrm{\Omega }}_{n}$ and any point D in spherical space ${S}_{n}\left(1\right)$, we establish an inequality for edge lengths of ${\mathrm{\Omega }}_{n}$ and distances from point D to faces of ${\mathrm{\Omega }}_{n}$, and from this we obtain some inequalities for the edge lengths and the in-radius of the simplex ${\mathrm{\Omega }}_{n}$. Besides, we establish some inequalities for the edge lengths and altitudes of a spherical simplex, and we establish inequalities for the edge lengths and circumradius of ${\mathrm{\Omega }}_{n}$.

MSC:51M10, 52A20, 51M20.

## 1 Introduction

The n-dimensional spherical space of curvature 1 is defined as follows (see [14]).

Let ${S}_{n}\left(1\right)=\left\{x\left({x}_{1},{x}_{2},\dots ,{x}_{n+1}\right)\mid {\sum }_{i=1}^{n+1}{x}_{i}^{2}=1\right\}$ be the n-dimensional unit sphere in the $\left(n+1\right)$-dimensional Euclidean ${E}^{n+1}$. For any two points $x\left({x}_{1},{x}_{2},\dots ,{x}_{n+1}\right)$, $y\left({y}_{1},{y}_{2},\dots ,{y}_{n+1}\right)\in {S}_{n}\left(1\right)$, the spherical distance between points x and y is defined as the smallest non-negative number $\stackrel{⁀}{xy}$ such that

$cos\stackrel{⁀}{xy}={x}_{1}{y}_{1}+{x}_{2}{y}_{2}+\cdots +{x}_{n+1}{y}_{n+1}.$

The n-dimensional unit sphere ${S}_{n}\left(1\right)$ with the above spherical distance is called the n-dimensional spherical space of curvature 1. Actually, the spherical space ${S}_{n}\left(1\right)$ is the boundary of an n-dimensional sphere of radius 1 in the $\left(n+1\right)$-dimensional Euclidean space ${E}^{n+1}$ with geodesic metric (that is, shorter arc).

Let ${\mathrm{\Omega }}_{n}$ be an n-dimensional simplex with vertices ${P}_{i}$ ($i=1,2,\dots ,n+1$) in the n-dimensional spherical space ${S}_{n}\left(1\right)$, r and R the in-radius and the circumradius of ${\mathrm{\Omega }}_{n}$, respectively. Let ${\rho }_{ij}=\stackrel{⁀}{{P}_{i}{P}_{j}}$ ($i\ne j$, $i,j=1,2,\dots ,n+1$) be the edge lengths of the spherical simplex ${\mathrm{\Omega }}_{n}$, ${h}_{i}$ the altitude of ${\mathrm{\Omega }}_{n}$ from vertex ${P}_{i}$, i.e. the spherical distance from point ${P}_{i}$ to the face ${f}_{i}=\left\{{P}_{1}\cdots {P}_{i-1}{P}_{i+1}\cdots {P}_{n+1}\right\}$ ($\left(n-1\right)$-dimensional spherical simplex) of ${\mathrm{\Omega }}_{n}$. Let D be any point inside the simplex ${\mathrm{\Omega }}_{n}$ and ${r}_{i}$ be the spherical distance from point D to the face ${f}_{i}$ of ${\mathrm{\Omega }}_{n}$ for $i=1,2,\dots ,n+1$.

For an n-simplex ${\mathrm{\Delta }}_{n}$ in the n-dimensional Euclidean space ${E}^{n}$, some important inequalities for the edge lengths of ${\mathrm{\Delta }}_{n}$ and ${r}_{i}$ ($i=1,2,\dots ,n+1$), inequalities for edge lengths and in-radius, circumradius, and altitudes of ${\mathrm{\Delta }}_{n}$ were established (see [510]). But similar inequalities for an n-simplex in the spherical space ${S}_{n}\left(1\right)$ have not been established. In this paper, we discuss the problems of inequalities for a spherical simplex and obtain some related inequalities for an n-simplex in the spherical space ${S}_{n}\left(1\right)$.

## 2 Inequalities for an n-simplex in the spherical space ${S}_{n}\left(1\right)$

In this section, we give some inequalities for the distances from an interior point to the faces of spherical simplex ${\mathrm{\Omega }}_{n}$ and inequalities for edge lengths and in-radius, circumradius, and altitudes of ${\mathrm{\Omega }}_{n}$. Our main results are the following theorems.

Let ${\phi }_{ij}$ ($i\ne j$, $i,j=1,2,\dots ,n+1$) be the dihedral angle formed by two faces ${f}_{i}$ and ${f}_{j}$ of an n-simplex ${\mathrm{\Omega }}_{n}$ in the spherical space ${S}_{n}\left(1\right)$.

Theorem 1 Let ${\mathrm{\Omega }}_{n}$ be an n-simplex in the n-dimensional spherical space ${S}_{n}\left(1\right)$ with dihedral angles ${\phi }_{ij}$ ($i\ne j$, $i,j=1,2,\dots ,n+1$), D be any interior point of simplex ${\mathrm{\Omega }}_{n}$ and ${r}_{i}$ the distance from the point D to the face ${f}_{i}$ of ${\mathrm{\Omega }}_{n}$ for $i=1,2,\dots ,n+1$. For any real numbers ${\lambda }_{i}\ne 0$ ($i=1,2,\dots ,n+1$), we have

$\begin{array}{rcl}\sum _{i=1}^{n+1}{\lambda }_{i}^{2}{cos}^{2}{r}_{i}& \le & \left[\frac{n}{2\left(n+1\right)}{\left(\sum _{i=1}^{n+1}{\lambda }_{i}^{2}+1\right)}^{2}-\sum _{1\le i
(1)

with equality if and only if the nonzero eigenvalues of matrix G are all equal. Here

$G=\left[\begin{array}{c}{\lambda }_{1}sin{r}_{1}\\ \overline{)-{\lambda }_{i}{\lambda }_{j}cos{\phi }_{ij}}& {\lambda }_{2}sin{r}_{2}\\ ⋮\\ {\lambda }_{n+1}sin{r}_{n+1}\\ {\lambda }_{1}sin{r}_{1}\phantom{\rule{1em}{0ex}}{\lambda }_{2}sin{r}_{2}\phantom{\rule{1em}{0ex}}\cdots \phantom{\rule{1em}{0ex}}{\lambda }_{n+1}sin{r}_{n+1}& 1\end{array}\right],$
(2)

and ${\phi }_{ii}=\pi$ ($i=1,2,\dots ,n+1$).

Let $M={\left(cos{\rho }_{ij}\right)}_{i,j=1}^{n+1}$ be the edge matrix of an n-simplex ${\mathrm{\Omega }}_{n}$ in ${S}_{n}\left(1\right)$, then M is a positive definite symmetric matrix with diagonal entries equal to 1 (see [3, 11]); by the cosine theorem of a simplex ${\mathrm{\Omega }}_{n}$ in ${S}_{n}\left(1\right)$ (see [13]), we have

$cos{\phi }_{ij}=-\frac{{M}_{ij}}{\sqrt{{M}_{ii}}\sqrt{{M}_{jj}}}\phantom{\rule{1em}{0ex}}\left(i,j=1,2,\dots ,n+1\right).$
(3)

Here ${M}_{ij}$ denotes the cofactor of matrix M corresponding to the $\left(i,j\right)$-entry. From Theorem 1 and (3) we get an inequality for ${r}_{i}$ ($i=1,2,\dots ,n+1$) and the edge lengths of spherical simplex ${\mathrm{\Omega }}_{n}$ as follows.

Theorem 1′ For any interior point D of an n-simplex ${\mathrm{\Omega }}_{n}$ in ${S}_{n}\left(1\right)$ and any real numbers ${\lambda }_{i}\ne 0$ ($i=1,2,\dots ,n+1$), we have

$\sum _{i=1}^{n+1}{\lambda }_{i}^{2}{cos}^{2}{r}_{i}\le \left[\frac{n}{2\left(n+1\right)}{\left(\sum _{i=1}^{n+1}{\lambda }_{i}^{2}+1\right)}^{2}-\sum _{1\le i
(4)

with equality if and only if the nonzero eigenvalues of matrix G are all equal.

If we take ${\lambda }_{i}^{2}={M}_{ii}$ ($i=1,2,\dots ,n+1$) in (4), we get the following corollary.

Corollary 1 For any interior point D of an n-simplex ${\mathrm{\Omega }}_{n}$ in ${S}_{n}\left(1\right)$, we have

$\sum _{i=1}^{n+1}{M}_{ii}{cos}^{2}{r}_{i}\le \left[\frac{n}{2\left(n+1\right)}{\left(\sum _{i=1}^{n+1}{M}_{ii}+1\right)}^{2}-\sum _{1\le i
(5)

Equality holds if and only if the nonzero eigenvalues of matrix G with ${\lambda }_{i}=\sqrt{{M}_{ii}}$ ($i=1,2,\dots ,n+1$) are all equal.

If we take the point D to be the in-center of ${\mathrm{\Omega }}_{n}$, then ${r}_{i}=r$ ($i=1,2,\dots ,n+1$) and from Theorem 1 and Theorem 1′, we get an inequality for the simplex ${\mathrm{\Omega }}_{n}$ as follows.

Corollary 2 For an n-simplex ${\mathrm{\Omega }}_{n}$ in ${S}_{n}\left(1\right)$ and real numbers ${\lambda }_{i}\ne 0$ ($i=1,2,\dots ,n+1$), we have

$\begin{array}{rcl}\left(\sum _{i=1}^{n+1}{\lambda }_{i}^{2}\right){cos}^{2}r& \le & \left[\frac{n}{2\left(n+1\right)}{\left(\sum _{i=1}^{n+1}{\lambda }_{i}^{2}+1\right)}^{2}-\sum _{1\le i
(6)

or

$\begin{array}{rcl}\left(\sum _{i=1}^{n+1}{\lambda }_{i}^{2}\right){cos}^{2}r& \le & \left[\frac{n}{2\left(n+1\right)}{\left(\sum _{i=1}^{n+1}{\lambda }_{i}^{2}+1\right)}^{2}-\sum _{1\le i
(7)

Equality holds if and only if the nonzero eigenvalues of matrix G with ${r}_{i}=r$ ($i=1,2,\dots ,n+1$) are all equal.

If we take ${\lambda }_{i}^{2}={M}_{ii}$ ($i=1,2,\dots ,n+1$) in (7), we get an inequality for the in-radius and the edge lengths of a simplex as follows.

Corollary 3 For an n-simplex ${\mathrm{\Omega }}_{n}$ in ${S}_{n}\left(1\right)$, we have

${cos}^{2}r\le \frac{1}{{\sum }_{i=1}^{n+1}{M}_{ii}}\left[\frac{n}{2\left(n+1\right)}{\left(\sum _{i=1}^{n+1}{M}_{ii}+1\right)}^{2}-\sum _{1\le i
(8)

Equality holds if and only if the nonzero eigenvalues of matrix G with ${r}_{i}=r$ and ${\lambda }_{i}=\sqrt{{M}_{ii}}$ ($i=1,2,\dots ,n+1$) are all equal.

Put ${\lambda }_{i}=1$ ($i=1,2,\dots ,n+1$) in (6) and (7), and we get the following corollary.

Corollary 4 For an n-simplex ${\mathrm{\Omega }}_{n}$ in ${S}_{n}\left(1\right)$, we have

${cos}^{2}r\le \frac{2{n}^{2}+3n}{2{\left(n+1\right)}^{2}}+\frac{1}{n+1}\sum _{1\le i
(9)

or

${cos}^{2}r\le \frac{2{n}^{2}+3n}{2{\left(n+1\right)}^{2}}+\frac{1}{n+1}\sum _{1\le i
(10)

Equality holds if and only if the nonzero eigenvalues of matrix G with ${r}_{i}=r$ and ${\lambda }_{i}=1$ ($i=1,2,\dots ,n+1$) are all equal.

Besides, we obtain an inequality for the edge lengths and circumradius of an n-simplex ${\mathrm{\Omega }}_{n}$ in ${S}_{n}\left(1\right)$ as follows.

Theorem 2 Let ${\rho }_{ij}$ ($i,j=1,2,\dots ,n+1$) and R be the edge lengths and the circumradius of an n-simplex ${\mathrm{\Omega }}_{n}$ in ${S}_{n}\left(1\right)$,respectively; let ${x}_{i}>0$ ($i=1,2,\dots ,n+1$) be real numbers, then we have

$\sum _{1\le i
(11)

Equality holds if and only if the nonzero eigenvalues of matrix B are all equal. Here

$B=\left[\begin{array}{c}\sqrt{{x}_{1}}cosR\\ \overline{)\sqrt{{x}_{i}{x}_{j}}cos{\rho }_{ij}}& ⋮\\ \sqrt{{x}_{n+1}}cosR\\ \sqrt{{x}_{1}}cosR\phantom{\rule{1em}{0ex}}\cdots \phantom{\rule{1em}{0ex}}\sqrt{{x}_{n+1}}cosR& 1\end{array}\right].$
(12)

If take ${x}_{1}={x}_{2}=\cdots ={x}_{n+1}=1$ in Theorem 2, we get an inequality as follows.

Corollary 5 For an n-simplex ${\mathrm{\Omega }}_{n}$ in ${S}_{n}\left(1\right)$, we have

$\sum _{1\le i
(13)

with equality holding if and only if the nonzero eigenvalues of matrix B with ${x}_{1}={x}_{2}=\cdots ={x}_{n+1}=1$ are all equal.

Finally, we give an inequality for edge lengths and altitudes of an n-simplex in ${S}_{n}\left(1\right)$ as follows.

Theorem 3 Let ${h}_{i}$ ($i=1,2,\dots ,n+1$) and M be the altitudes and the edge matrix of an n-simplex ${\mathrm{\Omega }}_{n}$ in ${S}_{n}\left(1\right)$, respectively; let ${x}_{i}>0$ ($i=1,2,\dots ,n+1$) be real numbers, then we have

$\sum _{i=1}^{n+1}\left(\prod _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{n+1}{x}_{j}\right){csc}^{2}{h}_{i}\ge \left(n+1\right){\left(\prod _{i=1}^{n+1}{x}_{i}\right)}^{\frac{n}{n+1}}\cdot {|M|}^{\frac{-1}{n+1}},$
(14)

with equality holding if and only if the eigenvalues of matrix Q are all equal. Here

$Q={\left(\sqrt{{x}_{i}{x}_{j}}cos{\rho }_{ij}\right)}_{i,j=1}^{n+1},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}M={\left(cos{\rho }_{ij}\right)}_{i,j=1}^{n+1}.$
(15)

If we take ${x}_{i}={csc}^{2}{h}_{i}$ ($i=1,2,\dots ,n+1$) in (14), we get the following corollary.

Corollary 6 For an n-simplex ${\mathrm{\Omega }}_{n}$ in ${S}_{n}\left(1\right)$, we have

$\prod _{i=1}^{n+1}sin{h}_{i}\le {|M|}^{\frac{1}{2}}\le {\left[\frac{2}{n\left(n+1\right)}\sum _{1\le i
(16)

with equality holding if ${\mathrm{\Omega }}_{n}$ is regular.

We will prove ${|M|}^{\frac{1}{2}}\le {\left[\frac{2}{n\left(n+1\right)}{\sum }_{1\le i and we have equality if ${\mathrm{\Omega }}_{n}$ is regular in the next section.

## 3 Proof of theorems

To prove the theorems in the above section, we need some lemmas.

Lemma 1 Let $M={\left(cos{\rho }_{ij}\right)}_{i,j=1}^{n+1}$ be the edge matrix of an n-simplex ${\mathrm{\Omega }}_{n}$ in ${S}_{n}\left(1\right)$, then M is a positive definite symmetric matrix with diagonal entries equal to 1.

For the proof of Lemma 1 one is referred to [3, 11].

Lemma 2 Let ${\phi }_{ij}$ be the dihedral angle formed by two faces ${f}_{i}$ and ${f}_{j}$ of an n-simplex ${\mathrm{\Omega }}_{n}$ in ${S}_{n}\left(1\right)$ for $i\ne j$, $i,j=1,2,\dots ,n+1$, and ${\phi }_{ii}=\pi$ ($i=1,2,\dots ,n+1$), then the Gram matrix $A={\left(-cos{\phi }_{ij}\right)}_{i,j=1}^{n+1}$ is positive definite symmetric matrix with diagonal entries equal to 1.

For the proof of Lemma 2 one is referred to [1].

Lemma 3 (see [12])

Let μ be the set of all points and oriented $\left(n-1\right)$-dimensional hyperplanes in the spherical space ${S}_{n}\left(1\right)$. For arbitrary m elements ${e}_{1},{e}_{2},\dots ,{e}_{m}$ of μ, define ${g}_{ij}$ as follows:

1. (i)

if ${e}_{i}$ and ${e}_{j}$ are two points, then ${g}_{ij}=cos\stackrel{⁀}{{e}_{i}{e}_{j}}$ (where $\stackrel{⁀}{{e}_{i}{e}_{j}}$ be spherical distance between ${e}_{i}$ and ${e}_{j}$);

2. (ii)

if ${e}_{i}$ and ${e}_{j}$ are unit outer normals of two unit outer normal of oriented , then ${g}_{ij}=cos\stackrel{ˆ}{{e}_{i}{e}_{j}}$ (where $\stackrel{ˆ}{{e}_{i}{e}_{j}}$ is dihedral angle formed by ${e}_{i}$ and ${e}_{j}$);

3. (iii)

if either of ${e}_{i}$ and ${e}_{j}$ is a point, and another is an outer normal, then ${g}_{ij}=sin{h}_{ij}$ (where ${h}_{ij}$ is the spherical distance with sign based on the direction from the point to the hyperplane).

If $m>n+1$, then

$det{\left({g}_{ij}\right)}_{i,j=1}^{m}=0.$

Lemma 4 Let ${h}_{i}$ be the altitude from vertex ${P}_{i}$ of an n-simplex ${\mathrm{\Omega }}_{n}$ in ${S}_{n}\left(1\right)$ for $i=1,2,\dots ,n+1$, and $M={\left(cos{\rho }_{ij}\right)}_{i,j=1}^{n+1}$ the edge matrix, then we have

${sin}^{2}{h}_{i}=\frac{|M|}{{M}_{ii}}\phantom{\rule{1em}{0ex}}\left(i=1,2,\dots ,n+1\right).$
(17)

For the proof of Lemma 4 one is referred to [13].

Proof of Theorem 1 Let ${e}_{i}$ is the unit outer normal of the oriented ${f}_{i}$ ($i=1,2,\dots ,n+1$) and the point ${e}_{n+2}=D$, such that $\stackrel{ˆ}{{e}_{i}{e}_{j}}=\pi -{\phi }_{ij}$ ($i,j=1,2,\dots ,n+1$) and the spherical distance with sign based on the direction from the point ${e}_{n+2}$ to the hyperplane ${e}_{i}$ is ${r}_{i}$ for $i=1,2,\dots ,n+1$.

By Lemma 2 we know that the $\left(n+1\right)×\left(n+1\right)$-order matrix ${\left(cos\stackrel{ˆ}{{e}_{i}{e}_{j}}\right)}_{i,j=1}^{n+1}={\left(-cos{\phi }_{ij}\right)}_{i,j=1}^{n+1}=A$ is a positive definite symmetric matrix. Because ${\lambda }_{i}\ne 0$ ($i=1,2,\dots ,n+1$), the matrix $T={\left(-{\lambda }_{i}{\lambda }_{j}cos{\phi }_{ij}\right)}_{i,j=1}^{n+1}$ is also a positive definite symmetric matrix.

By Lemma 3 we have

$B=|\begin{array}{c}sin{r}_{1}\\ \overline{)-cos{\phi }_{ij}}& ⋮\\ sin{r}_{n+1}\\ sin{r}_{1}\phantom{\rule{1em}{0ex}}\cdots \phantom{\rule{1em}{0ex}}sin{r}_{n+1}& 1\end{array}|=0.$
(18)

From (18) and ${\lambda }_{i}\ne 0$ ($i=1,2,\dots ,n+1$), we get

$detG=|\begin{array}{c}{\lambda }_{1}sin{r}_{1}\\ \overline{)-{\lambda }_{i}{\lambda }_{j}cos{\phi }_{ij}}& ⋮\\ {\lambda }_{n+1}sin{r}_{n+1}\\ {\lambda }_{1}sin{r}_{1}\phantom{\rule{1em}{0ex}}\cdots \phantom{\rule{1em}{0ex}}{\lambda }_{n+1}sin{r}_{n+1}& 1\end{array}|=0.$
(19)

Because the matrix $T={\left(-{\lambda }_{i}{\lambda }_{j}cos{\phi }_{ij}\right)}_{i,j=1}^{n+1}$ is also a positive definite symmetric matrix and $detG=0$, the matrix G is a semi-positive definite symmetric matrix and the rank of matrix G is $n+1$. Let ${u}_{i}>0$ ($i=1,2,\dots ,n+1$) and ${u}_{n+2}=0$ be the eigenvalues of the matrix G, and

${\sigma }_{1}=\sum _{i=1}^{n+2}{u}_{i}=\sum _{i=1}^{n+1}{u}_{i},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\sigma }_{2}=\sum _{1\le i

Using Maclaurin’s inequality [5], we have

${\left(\frac{1}{n+1}{\sigma }_{1}\right)}^{2}\ge \frac{2}{n\left(n+1\right)}{\sigma }_{2}.$
(20)

Equality holds if and only if ${u}_{1}={u}_{2}=\cdots ={u}_{n+1}$.

By the relation between the eigenvalues and the principal minors of the matrix G, we have

${\sigma }_{1}=\sum _{i=1}^{n+1}{\lambda }_{i}^{2}+1,\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\sigma }_{2}=\sum _{1\le i
(21)

Substituting (21) into (20), we get inequality (1). It is easy to see that equality holds in (1) if and only if the nonzero eigenvalues of matrix G are all equal. □

Proof of Theorem 2 Let C be the circumcenter of ${\mathrm{\Omega }}_{n}$, then $\stackrel{⁀}{C{P}_{i}}=R$ ($i=1,2,\dots ,n+1$). For real numbers ${x}_{i}>0$ ($i=1,2,\dots ,n+1$), by Lemma 1 we know that the matrix Q in (15) is a positive definite symmetric matrix. We take points ${e}_{i}={P}_{i}$ ($i=1,2,\dots ,n+1$) and ${e}_{n+2}=C$, and by Lemma 3 we have

$|\begin{array}{c}cosR\\ \overline{)cos{\rho }_{ij}}& ⋮\\ cosR\\ cosR\phantom{\rule{1em}{0ex}}\cdots \phantom{\rule{1em}{0ex}}cosR& 1\end{array}|=0.$

From this and ${x}_{i}>0$ ($i=1,2,\dots ,n+1$), we get

$detB=|\begin{array}{c}\sqrt{{x}_{1}}cosR\\ \overline{)\sqrt{{x}_{i}{x}_{j}}cos{\rho }_{ij}}& ⋮\\ \sqrt{{x}_{n+1}}cosR\\ \sqrt{{x}_{1}}cosR\phantom{\rule{1em}{0ex}}\cdots \phantom{\rule{1em}{0ex}}\sqrt{{x}_{n+1}}cosR& 1\end{array}|=0.$
(22)

Because the matrix $Q={\left(\sqrt{{x}_{i}{x}_{j}}cos{\rho }_{ij}\right)}_{i,j=1}^{n+1}$ is positive definite symmetric and $detB=0$, the matrix B is a semi-positive definite symmetric matrix and its rank is $n+1$. Let ${v}_{i}>0$ ($i=1,2,\dots ,n+1$), ${v}_{n+2}=0$ be the eigenvalues of matrix B, and

${\sigma }_{1}=\sum _{i=1}^{n+2}{v}_{i}=\sum _{i=1}^{n+1}{v}_{i},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\sigma }_{2}=\sum _{1\le i

Using Maclaurin’s inequality [5], we have

${\left(\frac{1}{n+1}{\sigma }_{1}\right)}^{2}\ge \frac{2}{n\left(n+1\right)}{\sigma }_{2}.$
(23)

Equality holds if and only if ${v}_{1}={v}_{2}=\cdots ={v}_{n+1}$.

By the relation between the eigenvalues and the principal minors of the matrix B, we have

${\sigma }_{1}=\sum _{i=1}^{n+1}{x}_{i}+1,\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\sigma }_{2}=\sum _{1\le i
(24)

Substituting (24) into (23), we get inequality (11). It is easy to see that equality holds in (11) if and only if the nonzero eigenvalues of matrix B are all equal. □

Proof of Theorem 3 From ${x}_{i}>0$ ($i=1,2,\dots ,n+1$) and the edge matrix $M={\left(cos{\rho }_{ij}\right)}_{i,j=1}^{n+1}$ of ${\mathrm{\Omega }}_{n}$ being a positive definite symmetric matrix, we know that the matrix Q in (15) is also a positive definite symmetric matrix. Let ${a}_{i}>0$ ($i=1,2,\dots ,n+1$) be the eigenvalues of the matrix Q, and

${\sigma }_{n}=\sum _{i=1}^{n+1}\underset{j\ne i}{\overset{n+1}{\prod _{j=1}}}{a}_{j},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\sigma }_{n+1}=\prod _{i=1}^{n+1}{a}_{i}.$

By Maclaurin’s inequality [5], we have

${\left(\frac{1}{n+1}{\sigma }_{n}\right)}^{\frac{1}{n}}\ge {\left({\sigma }_{n+1}\right)}^{\frac{1}{n+1}}.$
(25)

Equality holds if and only if ${a}_{1}={a}_{2}=\cdots ={a}_{n+1}$.

By the relation between the eigenvalues and the principal minors of the matrix Q, we have

${\sigma }_{n}=\sum _{i=1}^{n+1}{Q}_{ii}=\sum _{i=1}^{n+1}\left(\underset{j\ne i}{\overset{n+1}{\prod _{j=1}}}{x}_{j}\right){M}_{ii}\phantom{\rule{1em}{0ex}}\left(i=1,2,\dots ,n+1\right),$
(26)
${\sigma }_{n+1}=|Q|=\left(\prod _{i=1}^{n+1}{x}_{i}\right)\cdot |M|.$
(27)

From (25), (26), and (27), we get

$\sum _{i=1}^{n+1}\left(\prod _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{n+1}{x}_{j}\right){M}_{ii}\ge \left(n+1\right){\left(\prod _{i=1}^{n+1}{x}_{i}\right)}^{\frac{n}{n+1}}\cdot {|M|}^{\frac{n}{n+1}}.$
(28)

By Lemma 4 we have

${M}_{ii}=|M|{csc}^{2}{h}_{i}\phantom{\rule{1em}{0ex}}\left(i=1,2,\dots ,n+1\right).$
(29)

Substituting (29) into (28), we get inequality (14). It is easy to see that equality holds in (14) if and only if the eigenvalues of matrix Q are all equal.

Finally, we prove that inequality (30) is valid:

${|M|}^{\frac{1}{2}}\le {\left[\frac{2}{n\left(n+1\right)}\sum _{1\le i
(30)

Let ${b}_{i}$ ($i=1,2,\dots ,n+1$) be the eigenvalues of the edge matrix $M={\left(cos{\rho }_{ij}\right)}_{i,j=1}^{n+1}$. Since the matrix M is a positive definite symmetric matrix, ${b}_{i}>0$. Let

${\sigma }_{2}=\sum _{1\le i

By Maclaurin’s inequality [5], we have

${\left(\frac{2}{n\left(n+1\right)}{\sigma }_{2}\right)}^{\frac{1}{2}}\ge {\left({\sigma }_{n+1}\right)}^{\frac{1}{n+1}}.$
(31)

Equality holds if and only if ${b}_{1}={b}_{2}=\cdots ={b}_{n+1}$.

By the relation between the eigenvalues and the principal minors of the matrix M, we have

${\sigma }_{2}=\sum _{1\le i
(32)

From (31) and (32), we get inequality (30). If ${\mathrm{\Omega }}_{n}$ is a regular simplex in ${S}_{n}\left(1\right)$, then ${\rho }_{ij}=\frac{\pi }{2}$ ($i\ne j$, $i,j=1,2,\dots ,n+1$), $|M|=1$ and ${M}_{ii}=1$ ($i=1,2,\dots ,n+1$). By (17) we have $sin{h}_{i}=1$ ($i=1,2,\dots ,n+1$); thus equality holds in (16) if ${\mathrm{\Omega }}_{n}$ is a regular simplex. □

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## Acknowledgements

The work was supported by the Doctoral Programs Foundation of Education Ministry of China (20113401110009) and the Foundation of Anhui higher school (KJ2013A220). We are grateful for the help.

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Shi-guo, Y., Wen, W. & Ge, B. Inequalities for an n-simplex in spherical space ${S}_{n}\left(1\right)$. J Inequal Appl 2014, 59 (2014). https://doi.org/10.1186/1029-242X-2014-59

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• DOI: https://doi.org/10.1186/1029-242X-2014-59

### Keywords

• spherical simplex
• edge lengths