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

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.

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 (ij, 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 } ((n1)-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 (ij, 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 (ij, 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 ij, 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 (n1)-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

detG= | 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 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,,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

detB= | 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 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,,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 (ij, 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. □

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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 (1). J Inequal Appl 2014, 59 (2014). https://doi.org/10.1186/1029-242X-2014-59

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Keywords

  • spherical simplex
  • edge lengths
  • in-radius
  • circumradius
  • altitudes