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Inequalities for an n-simplex in spherical space
Journal of Inequalities and Applications volume 2014, Article number: 59 (2014)
Abstract
For an n-dimensional simplex and any point D in spherical space , we establish an inequality for edge lengths of and distances from point D to faces of , and from this we obtain some inequalities for the edge lengths and the in-radius of the simplex . 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 .
MSC:51M10, 52A20, 51M20.
1 Introduction
The n-dimensional spherical space of curvature 1 is defined as follows (see [1–4]).
Let be the n-dimensional unit sphere in the -dimensional Euclidean . For any two points , , the spherical distance between points x and y is defined as the smallest non-negative number such that
The n-dimensional unit sphere with the above spherical distance is called the n-dimensional spherical space of curvature 1. Actually, the spherical space is the boundary of an n-dimensional sphere of radius 1 in the -dimensional Euclidean space with geodesic metric (that is, shorter arc).
Let be an n-dimensional simplex with vertices () in the n-dimensional spherical space , r and R the in-radius and the circumradius of , respectively. Let (, ) be the edge lengths of the spherical simplex , the altitude of from vertex , i.e. the spherical distance from point to the face (-dimensional spherical simplex) of . Let D be any point inside the simplex and be the spherical distance from point D to the face of for .
For an n-simplex in the n-dimensional Euclidean space , some important inequalities for the edge lengths of and (), inequalities for edge lengths and in-radius, circumradius, and altitudes of were established (see [5–10]). But similar inequalities for an n-simplex in the spherical space 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 .
2 Inequalities for an n-simplex in the spherical space
In this section, we give some inequalities for the distances from an interior point to the faces of spherical simplex and inequalities for edge lengths and in-radius, circumradius, and altitudes of . Our main results are the following theorems.
Let (, ) be the dihedral angle formed by two faces and of an n-simplex in the spherical space .
Theorem 1 Let be an n-simplex in the n-dimensional spherical space with dihedral angles (, ), D be any interior point of simplex and the distance from the point D to the face of for . For any real numbers (), we have
with equality if and only if the nonzero eigenvalues of matrix G are all equal. Here
and ().
Let be the edge matrix of an n-simplex in , then M is a positive definite symmetric matrix with diagonal entries equal to 1 (see [3, 11]); by the cosine theorem of a simplex in (see [13]), we have
Here denotes the cofactor of matrix M corresponding to the -entry. From Theorem 1 and (3) we get an inequality for () and the edge lengths of spherical simplex as follows.
Theorem 1′ For any interior point D of an n-simplex in and any real numbers (), we have
with equality if and only if the nonzero eigenvalues of matrix G are all equal.
If we take () in (4), we get the following corollary.
Corollary 1 For any interior point D of an n-simplex in , we have
Equality holds if and only if the nonzero eigenvalues of matrix G with () are all equal.
If we take the point D to be the in-center of , then () and from Theorem 1 and Theorem 1′, we get an inequality for the simplex as follows.
Corollary 2 For an n-simplex in and real numbers (), we have
or
Equality holds if and only if the nonzero eigenvalues of matrix G with () are all equal.
If we take () 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 in , we have
Equality holds if and only if the nonzero eigenvalues of matrix G with and () are all equal.
Put () in (6) and (7), and we get the following corollary.
Corollary 4 For an n-simplex in , we have
or
Equality holds if and only if the nonzero eigenvalues of matrix G with and () are all equal.
Besides, we obtain an inequality for the edge lengths and circumradius of an n-simplex in as follows.
Theorem 2 Let () and R be the edge lengths and the circumradius of an n-simplex in ,respectively; let () be real numbers, then we have
Equality holds if and only if the nonzero eigenvalues of matrix B are all equal. Here
If take in Theorem 2, we get an inequality as follows.
Corollary 5 For an n-simplex in , we have
with equality holding if and only if the nonzero eigenvalues of matrix B with are all equal.
Finally, we give an inequality for edge lengths and altitudes of an n-simplex in as follows.
Theorem 3 Let () and M be the altitudes and the edge matrix of an n-simplex in , respectively; let () be real numbers, then we have
with equality holding if and only if the eigenvalues of matrix Q are all equal. Here
If we take () in (14), we get the following corollary.
Corollary 6 For an n-simplex in , we have
with equality holding if is regular.
We will prove and we have equality if 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 be the edge matrix of an n-simplex in , 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 be the dihedral angle formed by two faces and of an n-simplex in for , , and (), then the Gram matrix 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 -dimensional hyperplanes in the spherical space . For arbitrary m elements of μ, define as follows:
-
(i)
if and are two points, then (where be spherical distance between and );
-
(ii)
if and are unit outer normals of two unit outer normal of oriented , then (where is dihedral angle formed by and );
-
(iii)
if either of and is a point, and another is an outer normal, then (where is the spherical distance with sign based on the direction from the point to the hyperplane).
If , then
Lemma 4 Let be the altitude from vertex of an n-simplex in for , and the edge matrix, then we have
For the proof of Lemma 4 one is referred to [13].
Proof of Theorem 1 Let is the unit outer normal of the oriented () and the point , such that () and the spherical distance with sign based on the direction from the point to the hyperplane is for .
By Lemma 2 we know that the -order matrix is a positive definite symmetric matrix. Because (), the matrix is also a positive definite symmetric matrix.
By Lemma 3 we have
From (18) and (), we get
Because the matrix is also a positive definite symmetric matrix and , the matrix G is a semi-positive definite symmetric matrix and the rank of matrix G is . Let () and be the eigenvalues of the matrix G, and
Using Maclaurin’s inequality [5], we have
Equality holds if and only if .
By the relation between the eigenvalues and the principal minors of the matrix G, we have
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 , then (). For real numbers (), by Lemma 1 we know that the matrix Q in (15) is a positive definite symmetric matrix. We take points () and , and by Lemma 3 we have
From this and (), we get
Because the matrix is positive definite symmetric and , the matrix B is a semi-positive definite symmetric matrix and its rank is . Let (), be the eigenvalues of matrix B, and
Using Maclaurin’s inequality [5], we have
Equality holds if and only if .
By the relation between the eigenvalues and the principal minors of the matrix B, we have
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 () and the edge matrix of being a positive definite symmetric matrix, we know that the matrix Q in (15) is also a positive definite symmetric matrix. Let () be the eigenvalues of the matrix Q, and
By Maclaurin’s inequality [5], we have
Equality holds if and only if .
By the relation between the eigenvalues and the principal minors of the matrix Q, we have
From (25), (26), and (27), we get
By Lemma 4 we have
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:
Let () be the eigenvalues of the edge matrix . Since the matrix M is a positive definite symmetric matrix, . Let
By Maclaurin’s inequality [5], we have
Equality holds if and only if .
By the relation between the eigenvalues and the principal minors of the matrix M, we have
From (31) and (32), we get inequality (30). If is a regular simplex in , then (, ), and (). By (17) we have (); thus equality holds in (16) if 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 . 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