Open Access

Inequalities for an n-simplex in spherical space S n ( 1 )

Journal of Inequalities and Applications20142014:59

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

Received: 26 August 2013

Accepted: 20 January 2014

Published: 10 February 2014

Abstract

For an n-dimensional simplex Ω n and any point D in spherical space S n ( 1 ) , we establish an inequality for edge lengths of Ω n and distances from point D to faces of Ω n , and from this we obtain some inequalities for the edge lengths and the in-radius of the simplex Ω 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 Ω n .

MSC:51M10, 52A20, 51M20.

Keywords

spherical simplex edge lengths in-radius circumradius altitudes

1 Introduction

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

Let S n ( 1 ) = { x ( x 1 , x 2 , , x n + 1 ) i = 1 n + 1 x i 2 = 1 } be the n-dimensional unit sphere in the ( n + 1 ) -dimensional Euclidean E n + 1 . For any two points x ( x 1 , x 2 , , x n + 1 ) , y ( y 1 , y 2 , , y n + 1 ) S n ( 1 ) , the spherical distance between points x and y is defined as the smallest non-negative number x y such that
cos x y = x 1 y 1 + x 2 y 2 + + x n + 1 y n + 1 .

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

Let Ω n be an n-dimensional simplex with vertices P i ( i = 1 , 2 , , n + 1 ) in the n-dimensional spherical space S n ( 1 ) , r and R the in-radius and the circumradius of Ω n , respectively. Let ρ i j = P i P j ( i j , i , j = 1 , 2 , , n + 1 ) be the edge lengths of the spherical simplex Ω n , h i the altitude of Ω n from vertex P i , i.e. the spherical distance from point P i to the face f i = { P 1 P i 1 P i + 1 P n + 1 } ( ( n 1 ) -dimensional spherical simplex) of Ω n . Let D be any point inside the simplex Ω n and r i be the spherical distance from point D to the face f i of Ω n for i = 1 , 2 , , n + 1 .

For an n-simplex Δ n in the n-dimensional Euclidean space E n , some important inequalities for the edge lengths of Δ n and r i ( i = 1 , 2 , , n + 1 ), inequalities for edge lengths and in-radius, circumradius, and altitudes of Δ n were established (see [510]). But similar inequalities for an n-simplex in the spherical space S n ( 1 ) 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 ( 1 ) .

2 Inequalities for an n-simplex in the spherical space S n ( 1 )

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

Let φ i j ( i j , i , j = 1 , 2 , , n + 1 ) be the dihedral angle formed by two faces f i and f j of an n-simplex Ω n in the spherical space S n ( 1 ) .

Theorem 1 Let Ω n be an n-simplex in the n-dimensional spherical space S n ( 1 ) with dihedral angles φ i j ( i j , i , j = 1 , 2 , , n + 1 ), D be any interior point of simplex Ω n and r i the distance from the point D to the face f i of Ω n for i = 1 , 2 , , n + 1 . For any real numbers λ i 0 ( i = 1 , 2 , , n + 1 ), we have
i = 1 n + 1 λ i 2 cos 2 r i [ n 2 ( n + 1 ) ( i = 1 n + 1 λ i 2 + 1 ) 2 1 i < j n + 1 λ i 2 λ j 2 ] + 1 i < j n + 1 λ i 2 λ j 2 cos 2 φ i j ,
(1)
with equality if and only if the nonzero eigenvalues of matrix G are all equal. Here
G = [ λ 1 sin r 1 λ i λ j cos φ i j λ 2 sin r 2 λ n + 1 sin r n + 1 λ 1 sin r 1 λ 2 sin r 2 λ n + 1 sin r n + 1 1 ] ,
(2)

and φ i i = π ( i = 1 , 2 , , n + 1 ).

Let M = ( cos ρ i j ) i , j = 1 n + 1 be the edge matrix of an n-simplex Ω n in S n ( 1 ) , then M is a positive definite symmetric matrix with diagonal entries equal to 1 (see [3, 11]); by the cosine theorem of a simplex Ω n in S n ( 1 ) (see [13]), we have
cos φ i j = M i j M i i M j j ( i , j = 1 , 2 , , n + 1 ) .
(3)

Here M i j denotes the cofactor of matrix M corresponding to the ( i , j ) -entry. From Theorem 1 and (3) we get an inequality for r i ( i = 1 , 2 , , n + 1 ) and the edge lengths of spherical simplex Ω n as follows.

Theorem 1′ For any interior point D of an n-simplex Ω n in S n ( 1 ) and any real numbers λ i 0 ( i = 1 , 2 , , n + 1 ), we have
i = 1 n + 1 λ i 2 cos 2 r i [ n 2 ( n + 1 ) ( i = 1 n + 1 λ i 2 + 1 ) 2 1 i < j n + 1 λ i 2 λ j 2 ] + 1 i < j n + 1 λ i 2 λ j 2 M i j 2 M i i M j j ,
(4)

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

If we take λ i 2 = M i i ( i = 1 , 2 , , n + 1 ) in (4), we get the following corollary.

Corollary 1 For any interior point D of an n-simplex Ω n in S n ( 1 ) , we have
i = 1 n + 1 M i i cos 2 r i [ n 2 ( n + 1 ) ( i = 1 n + 1 M i i + 1 ) 2 1 i < j n + 1 M i i M j j ] + 1 i < j n + 1 M i j 2 .
(5)

Equality holds if and only if the nonzero eigenvalues of matrix G with λ i = M i i ( i = 1 , 2 , , n + 1 ) are all equal.

If we take the point D to be the in-center of Ω n , then r i = r ( i = 1 , 2 , , n + 1 ) and from Theorem 1 and Theorem 1′, we get an inequality for the simplex Ω n as follows.

Corollary 2 For an n-simplex Ω n in S n ( 1 ) and real numbers λ i 0 ( i = 1 , 2 , , n + 1 ), we have
( i = 1 n + 1 λ i 2 ) cos 2 r [ n 2 ( n + 1 ) ( i = 1 n + 1 λ i 2 + 1 ) 2 1 i < j n + 1 λ i 2 λ j 2 ] + 1 i < j n + 1 λ i 2 λ j 2 cos 2 φ i j ,
(6)
or
( i = 1 n + 1 λ i 2 ) cos 2 r [ n 2 ( n + 1 ) ( i = 1 n + 1 λ i 2 + 1 ) 2 1 i < j n + 1 λ i 2 λ j 2 ] + 1 i < j n + 1 λ i 2 λ j 2 M i j 2 M i i M j j .
(7)

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

If we take λ i 2 = M i i ( i = 1 , 2 , , 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 Ω n in S n ( 1 ) , we have
cos 2 r 1 i = 1 n + 1 M i i [ n 2 ( n + 1 ) ( i = 1 n + 1 M i i + 1 ) 2 1 i < j n + 1 M i i M j j + 1 i < j n + 1 M i j 2 ] .
(8)

Equality holds if and only if the nonzero eigenvalues of matrix G with r i = r and λ i = M i i ( i = 1 , 2 , , n + 1 ) are all equal.

Put λ i = 1 ( i = 1 , 2 , , n + 1 ) in (6) and (7), and we get the following corollary.

Corollary 4 For an n-simplex Ω n in S n ( 1 ) , we have
cos 2 r 2 n 2 + 3 n 2 ( n + 1 ) 2 + 1 n + 1 1 i < j n + 1 M i j 2 M i i M j j ,
(9)
or
cos 2 r 2 n 2 + 3 n 2 ( n + 1 ) 2 + 1 n + 1 1 i < j n + 1 cos 2 φ i j .
(10)

Equality holds if and only if the nonzero eigenvalues of matrix G with r i = r and λ i = 1 ( i = 1 , 2 , , n + 1 ) are all equal.

Besides, we obtain an inequality for the edge lengths and circumradius of an n-simplex Ω n in S n ( 1 ) as follows.

Theorem 2 Let ρ i j ( i , j = 1 , 2 , , n + 1 ) and R be the edge lengths and the circumradius of an n-simplex Ω n in S n ( 1 ) ,respectively; let x i > 0 ( i = 1 , 2 , , n + 1 ) be real numbers, then we have
1 i < j n + 1 x i x j sin 2 ρ i j [ n 2 ( n + 1 ) ( i = 1 n + 1 x i + 1 ) 2 i = 1 n + 1 x i ] + ( i = 1 n + 1 x i ) cos 2 R .
(11)
Equality holds if and only if the nonzero eigenvalues of matrix B are all equal. Here
B = [ x 1 cos R x i x j cos ρ i j x n + 1 cos R x 1 cos R x n + 1 cos R 1 ] .
(12)

If take x 1 = x 2 = = x n + 1 = 1 in Theorem 2, we get an inequality as follows.

Corollary 5 For an n-simplex Ω n in S n ( 1 ) , we have
1 i < j n + 1 sin 2 ρ i j n 3 + 2 n 2 2 2 ( n + 1 ) + ( n + 1 ) cos 2 R ,
(13)

with equality holding if and only if the nonzero eigenvalues of matrix B with x 1 = x 2 = = x n + 1 = 1 are all equal.

Finally, we give an inequality for edge lengths and altitudes of an n-simplex in S n ( 1 ) as follows.

Theorem 3 Let h i ( i = 1 , 2 , , n + 1 ) and M be the altitudes and the edge matrix of an n-simplex Ω n in S n ( 1 ) , respectively; let x i > 0 ( i = 1 , 2 , , n + 1 ) be real numbers, then we have
i = 1 n + 1 ( j = 1 j i n + 1 x j ) csc 2 h i ( n + 1 ) ( i = 1 n + 1 x i ) n n + 1 | M | 1 n + 1 ,
(14)
with equality holding if and only if the eigenvalues of matrix Q are all equal. Here
Q = ( x i x j cos ρ i j ) i , j = 1 n + 1 , M = ( cos ρ i j ) i , j = 1 n + 1 .
(15)

If we take x i = csc 2 h i ( i = 1 , 2 , , n + 1 ) in (14), we get the following corollary.

Corollary 6 For an n-simplex Ω n in S n ( 1 ) , we have
i = 1 n + 1 sin h i | M | 1 2 [ 2 n ( n + 1 ) 1 i < j n + 1 sin 2 ρ i j ] n + 1 4 ,
(16)

with equality holding if Ω n is regular.

We will prove | M | 1 2 [ 2 n ( n + 1 ) 1 i < j n + 1 sin 2 ρ i j ] n + 1 4 and we have equality if Ω 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 = ( cos ρ i j ) i , j = 1 n + 1 be the edge matrix of an n-simplex Ω n in S n ( 1 ) , 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 φ i j be the dihedral angle formed by two faces f i and f j of an n-simplex Ω n in S n ( 1 ) for i j , i , j = 1 , 2 , , n + 1 , and φ i i = π ( i = 1 , 2 , , n + 1 ), then the Gram matrix A = ( cos φ i j ) 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 ( n 1 ) -dimensional hyperplanes in the spherical space S n ( 1 ) . For arbitrary m elements e 1 , e 2 , , e m of μ, define g i j as follows:
  1. (i)

    if e i and e j are two points, then g i j = cos e i e j (where 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 i j = cos e i e j ˆ (where e i e j ˆ 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 i j = sin h i j (where h i j is the spherical distance with sign based on the direction from the point to the hyperplane).

     
If m > n + 1 , then
det ( g i j ) i , j = 1 m = 0 .
Lemma 4 Let h i be the altitude from vertex P i of an n-simplex Ω n in S n ( 1 ) for i = 1 , 2 , , n + 1 , and M = ( cos ρ i j ) i , j = 1 n + 1 the edge matrix, then we have
sin 2 h i = | M | M i i ( i = 1 , 2 , , n + 1 ) .
(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 , , n + 1 ) and the point e n + 2 = D , such that e i e j ˆ = π φ i j ( i , j = 1 , 2 , , 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 , , n + 1 .

By Lemma 2 we know that the ( n + 1 ) × ( n + 1 ) -order matrix ( cos e i e j ˆ ) i , j = 1 n + 1 = ( cos φ i j ) i , j = 1 n + 1 = A is a positive definite symmetric matrix. Because λ i 0 ( i = 1 , 2 , , n + 1 ), the matrix T = ( λ i λ j cos φ i j ) i , j = 1 n + 1 is also a positive definite symmetric matrix.

By Lemma 3 we have
B = | sin r 1 cos φ i j sin r n + 1 sin r 1 sin r n + 1 1 | = 0 .
(18)
From (18) and λ i 0 ( i = 1 , 2 , , n + 1 ), we get
det G = | λ 1 sin r 1 λ i λ j cos φ i j λ n + 1 sin r n + 1 λ 1 sin r 1 λ n + 1 sin r n + 1 1 | = 0 .
(19)
Because the matrix T = ( λ i λ j cos φ i j ) i , j = 1 n + 1 is also a positive definite symmetric matrix and det G = 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 , , n + 1 ) and u n + 2 = 0 be the eigenvalues of the matrix G, and
σ 1 = i = 1 n + 2 u i = i = 1 n + 1 u i , σ 2 = 1 i < j n + 2 u i u j = 1 i < j n + 1 u i u j .
Using Maclaurin’s inequality [5], we have
( 1 n + 1 σ 1 ) 2 2 n ( n + 1 ) σ 2 .
(20)

Equality holds if and only if u 1 = u 2 = = u n + 1 .

By the relation between the eigenvalues and the principal minors of the matrix G, we have
σ 1 = i = 1 n + 1 λ i 2 + 1 , σ 2 = 1 i < j n + 1 λ i 2 λ j 2 sin 2 φ i j + i = 1 n + 1 λ i 2 cos 2 r 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 Ω n , then C P i = R ( i = 1 , 2 , , n + 1 ). For real numbers x i > 0 ( i = 1 , 2 , , 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 , , n + 1 ) and e n + 2 = C , and by Lemma 3 we have
| cos R cos ρ i j cos R cos R cos R 1 | = 0 .
From this and x i > 0 ( i = 1 , 2 , , n + 1 ), we get
det B = | x 1 cos R x i x j cos ρ i j x n + 1 cos R x 1 cos R x n + 1 cos R 1 | = 0 .
(22)
Because the matrix Q = ( x i x j cos ρ i j ) i , j = 1 n + 1 is positive definite symmetric and det B = 0 , the matrix B is a semi-positive definite symmetric matrix and its rank is n + 1 . Let v i > 0 ( i = 1 , 2 , , n + 1 ), v n + 2 = 0 be the eigenvalues of matrix B, and
σ 1 = i = 1 n + 2 v i = i = 1 n + 1 v i , σ 2 = 1 i < j n + 2 v i v j = 1 i < j n + 1 v i v j .
Using Maclaurin’s inequality [5], we have
( 1 n + 1 σ 1 ) 2 2 n ( n + 1 ) σ 2 .
(23)

Equality holds if and only if v 1 = v 2 = = v n + 1 .

By the relation between the eigenvalues and the principal minors of the matrix B, we have
σ 1 = i = 1 n + 1 x i + 1 , σ 2 = 1 i < j n + 1 x i x j sin 2 ρ i j + i = 1 n + 1 x i ( 1 cos 2 R ) .
(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 , , n + 1 ) and the edge matrix M = ( cos ρ i j ) i , j = 1 n + 1 of  Ω 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 , , n + 1 ) be the eigenvalues of the matrix Q, and
σ n = i = 1 n + 1 j = 1 j i n + 1 a j , σ n + 1 = i = 1 n + 1 a i .
By Maclaurin’s inequality [5], we have
( 1 n + 1 σ n ) 1 n ( σ n + 1 ) 1 n + 1 .
(25)

Equality holds if and only if a 1 = a 2 = = a n + 1 .

By the relation between the eigenvalues and the principal minors of the matrix Q, we have
σ n = i = 1 n + 1 Q i i = i = 1 n + 1 ( j = 1 j i n + 1 x j ) M i i ( i = 1 , 2 , , n + 1 ) ,
(26)
σ n + 1 = | Q | = ( i = 1 n + 1 x i ) | M | .
(27)
From (25), (26), and (27), we get
i = 1 n + 1 ( j = 1 j i n + 1 x j ) M i i ( n + 1 ) ( i = 1 n + 1 x i ) n n + 1 | M | n n + 1 .
(28)
By Lemma 4 we have
M i i = | M | csc 2 h i ( i = 1 , 2 , , n + 1 ) .
(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 | 1 2 [ 2 n ( n + 1 ) 1 i < j n + 1 sin 2 ρ i j ] n + 1 4 .
(30)
Let b i ( i = 1 , 2 , , n + 1 ) be the eigenvalues of the edge matrix M = ( cos ρ i j ) i , j = 1 n + 1 . Since the matrix M is a positive definite symmetric matrix, b i > 0 . Let
σ 2 = 1 i < j n + 1 b i b j , σ n + 1 = i = 1 n + 1 b i .
By Maclaurin’s inequality [5], we have
( 2 n ( n + 1 ) σ 2 ) 1 2 ( σ n + 1 ) 1 n + 1 .
(31)

Equality holds if and only if b 1 = b 2 = = b n + 1 .

By the relation between the eigenvalues and the principal minors of the matrix M, we have
σ 2 = 1 i < j n + 1 sin 2 ρ i j , σ n + 1 = | M | .
(32)

From (31) and (32), we get inequality (30). If Ω n is a regular simplex in S n ( 1 ) , then ρ i j = π 2 ( i j , i , j = 1 , 2 , , n + 1 ), | M | = 1 and M i i = 1 ( i = 1 , 2 , , n + 1 ). By (17) we have sin h i = 1 ( i = 1 , 2 , , n + 1 ); thus equality holds in (16) if Ω n is a regular simplex. □

Declarations

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.

Authors’ Affiliations

(1)
Department of Mathematics and Teachers Educational Research, Hefei Normal University
(2)
Anhui Xinhua University
(3)
School of mathematical Since, Anhui University

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