R E S E A R C H
Open Access
Inequalities for an
n-simplex in spherical
space
S
n
()
Yang Shi-guo
1,2,3*, Wang Wen
1*and Bian Ge
3*Correspondence:
yangsg@hftc.edu.cn; wangwen0811@163.com
1Department of Mathematics and
Teachers Educational Research, Hefei Normal University, Hefei, 230601, P.R. China
2Anhui Xinhua University, Hefei,
230088, P.R. China
Full list of author information is available at the end of the article
Abstract
For ann-dimensional simplex
nand any pointDin spherical spaceSn(1), we
establish an inequality for edge lengths of
nand distances from pointDto faces of
n, and from this we obtain some inequalities for the edge lengths and the in-radius
of the simplex
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
n.
MSC: 51M10; 52A20; 51M20
Keywords: spherical simplex; edge lengths; in-radius; circumradius; altitudes
1 Introduction
Then-dimensional spherical space of curvature is defined as follows (see [–]). Let Sn() ={x(x,x, . . . ,xn+)|n+i= xi = } be the n-dimensional unit sphere in the (n+ )-dimensional EuclideanEn+. For any two pointsx(x,x, . . . ,xn+),y(y,y, . . . ,yn+)∈ Sn(), the spherical distance between pointsxandyis defined as the smallest non-negative numberxyÙsuch that
cosxyÙ=xy+xy+· · ·+xn+yn+.
Then-dimensional unit sphereSn() with the above spherical distance is called the n-dimensional spherical space of curvature . Actually, the spherical spaceSn() is the bound-ary of ann-dimensional sphere of radius in the (n+ )-dimensional Euclidean spaceEn+ with geodesic metric (that is, shorter arc).
Let n be an n-dimensional simplex with vertices Pi (i = , , . . . ,n+ ) in the n-dimensional spherical spaceSn(),randRthe in-radius and the circumradius ofn, re-spectively. Letρij=P˜iPj(i=j,i,j= , , . . . ,n+ ) be the edge lengths of the spherical sim-plexn,hithe altitude ofnfrom vertexPi,i.e.the spherical distance from pointPi to the facefi={P· · ·Pi–Pi+· · ·Pn+}((n– )-dimensional spherical simplex) ofn. LetDbe any point inside the simplexnandribe the spherical distance from pointDto the face fiofnfori= , , . . . ,n+ .
For ann-simplexnin then-dimensional Euclidean spaceEn, some important inequal-ities for the edge lengths ofnandri(i= , , . . . ,n+ ), inequalities for edge lengths and in-radius, circumradius, and altitudes ofnwere established (see [–]). But similar in-equalities for ann-simplex in the spherical spaceSn() have not been established. In this
paper, we discuss the problems of inequalities for a spherical simplex and obtain some related inequalities for ann-simplex in the spherical spaceSn().
2 Inequalities for ann-simplex in the spherical spaceSn(1)
In this section, we give some inequalities for the distances from an interior point to the faces of spherical simplexnand inequalities for edge lengths and in-radius, circumra-dius, and altitudes ofn. Our main results are the following theorems.
Letϕij(i=j,i,j= , , . . . ,n+ ) be the dihedral angle formed by two facesfiandfjof an n-simplexnin the spherical spaceSn().
Theorem Letnbe an n-simplex in the n-dimensional spherical space Sn()with dihe-dral anglesϕij(i=j,i,j= , , . . . ,n+ ),D be any interior point of simplexnand rithe distance from the point D to the face fi ofnfor i= , , . . . ,n+ .For any real numbers
λi= (i= , , . . . ,n+ ),we have n+
i=
λicosri≤
ñ
n (n+ )
Çn+
i=
λi +
å
–
≤i<j≤n+
λiλj
ô
+
≤i<j≤n+
λiλjcosϕij, ()
with equality if and only if the nonzero eigenvalues of matrix G are all equal.Here
G=
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
λsinr –λiλjcosϕij λsinr
.. .
λn+sinrn+
λsinr λsinr · · · λn+sinrn+ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
, ()
andϕii=π(i= , , . . . ,n+ ).
LetM= (cosρij)n+
i,j= be the edge matrix of ann-simplexninSn(), thenMis a posi-tive definite symmetric matrix with diagonal entries equal to (see [, ]); by the cosine theorem of a simplexninSn() (see []), we have
cosϕij= – Mij
√
Mii
Mjj
(i,j= , , . . . ,n+ ). ()
HereMijdenotes the cofactor of matrixMcorresponding to the (i,j)-entry. From Theo-rem and () we get an inequality forri(i= , , . . . ,n+ ) and the edge lengths of spherical simplexnas follows.
Theorem For any interior point D of an n-simplexnin Sn()and any real numbers
λi= (i= , , . . . ,n+ ),we have n+
i=
λicosri≤
ñ
n (n+ )
Çn+
i=
λi +
å
–
≤i<j≤n+
λiλj
ô
+
≤i<j≤n+
λiλj M
ij MiiMjj
, ()
If we takeλ
i =Mii(i= , , . . . ,n+ ) in (), we get the following corollary.
Corollary For any interior point D of an n-simplexnin Sn(),we have
n+
i=
Miicosri≤
ñ
n (n+ )
Çn+
i= Mii+
å
–
≤i<j≤n+ MiiMjj
ô
+
≤i<j≤n+
Mij. ()
Equality holds if and only if the nonzero eigenvalues of matrix G with λi=
√
Mii (i= , , . . . ,n+ )are all equal.
If we take the pointDto be the in-center ofn, thenri=r(i= , , . . . ,n+ ) and from Theorem and Theorem , we get an inequality for the simplexnas follows.
Corollary For an n-simplexnin Sn()and real numbersλi= (i= , , . . . ,n+ ),we have
Çn+
i=
λi
å cosr≤
ñ
n (n+ )
Çn+
i=
λi +
å
–
≤i<j≤n+
λiλj
ô
+
≤i<j≤n+
λiλjcosϕij, ()
or
Çn+
i=
λi
å cosr≤
ñ
n (n+ )
Çn+
i=
λi +
å
–
≤i<j≤n+
λiλj
ô
+
≤i<j≤n+
λiλj M
ij
MiiMjj. ()
Equality holds if and only if the nonzero eigenvalues of matrix G with ri=r(i= , , . . . ,n+) are all equal.
If we takeλ
i =Mii(i= , , . . . ,n+ ) in (), we get an inequality for the in-radius and the edge lengths of a simplex as follows.
Corollary For an n-simplexnin Sn(),we have
cosr≤n+ i=Mii
ñ
n (n+ )
Çn+
i= Mii+
å
–
≤i<j≤n+
MiiMjj+
≤i<j≤n+ Mij
ô
. ()
Equality holds if and only if the nonzero eigenvalues of matrix G with ri=r andλi=
√
Mii (i= , , . . . ,n+ )are all equal.
Putλi= (i= , , . . . ,n+ ) in () and (), and we get the following corollary.
Corollary For an n-simplexnin Sn(),we have
cosr≤n + n
(n+ ) + n+
≤i<j≤n+ Mij
or
cosr≤n + n
(n+ ) + n+
≤i<j≤n+
cosϕij. ()
Equality holds if and only if the nonzero eigenvalues of matrix G with ri=r andλi= (i= , , . . . ,n+ )are all equal.
Besides, we obtain an inequality for the edge lengths and circumradius of ann-simplex
ninSn() as follows.
Theorem Letρij(i,j= , , . . . ,n+ )and R be the edge lengths and the circumradius of an n-simplexnin Sn(),respectively;let xi> (i= , , . . . ,n+ )be real numbers,then we have
≤i<j≤n+
xixjsinρij≤
ñ
n (n+ )
Çn+
i= xi+
å –
n+
i= xi
ô
+
Çn+
i= xi
å
cosR. ()
Equality holds if and only if the nonzero eigenvalues of matrix B are all equal.Here
B= ⎡ ⎢ ⎢ ⎢ ⎢ ⎣
√
xcosR
√
xixjcosρij ...
√
xn+cosR
√
xcosR · · · √xn+cosR ⎤ ⎥ ⎥ ⎥ ⎥
⎦. ()
If takex=x=· · ·=xn+= in Theorem , we get an inequality as follows.
Corollary For an n-simplexnin Sn(),we have
≤i<j≤n+
sinρij≤
n+ n–
(n+ ) + (n+ )cos
R, ()
with equality holding if and only if the nonzero eigenvalues of matrix B with x=x=· · ·= xn+= are all equal.
Finally, we give an inequality for edge lengths and altitudes of ann-simplex inSn() as follows.
Theorem Let hi(i= , , . . . ,n+ )and M be the altitudes and the edge matrix of an n-simplexnin Sn(),respectively;let xi> (i= , , . . . ,n+ )be real numbers,then we have
n+
i=
Çn+
j= j=i
xj
å
cschi≥(n+ )
Çn+
i= xi
å n n+
· |M|n+–, ()
with equality holding if and only if the eigenvalues of matrix Q are all equal.Here
If we takexi=cschi(i= , , . . . ,n+ ) in (), we get the following corollary.
Corollary For an n-simplexnin Sn(),we have
n+
i=
sinhi≤ |M|≤
ï
n(n+ )
≤i<j≤n+
sinρij
òn+
, ()
with equality holding ifnis regular.
We will prove|M|≤[
n(n+)
≤i<j≤n+sinρij]
n+
and we have equality ifnis regular
in the next section.
3 Proof of theorems
To prove the theorems in the above section, we need some lemmas.
Lemma Let M= (cosρij)n+
i,j=be the edge matrix of an n-simplexnin Sn(),then M is a positive definite symmetric matrix with diagonal entries equal to.
For the proof of Lemma one is referred to [, ].
Lemma Letϕijbe the dihedral angle formed by two faces fiand fjof an n-simplexn in Sn()for i=j,i,j= , , . . . ,n+ ,andϕii=π(i= , , . . . ,n+ ),then the Gram matrix A= (–cosϕij)n+i,j=is positive definite symmetric matrix with diagonal entries equal to.
For the proof of Lemma one is referred to [].
Lemma (see []) Letμbe the set of all points and oriented(n– )-dimensional hyper-planes in the spherical space Sn().For arbitrary m elements e,e, . . . ,emofμ,define gijas follows:
(i) ifeiandejare two points,thengij=cosˆeiej(whereeiejˆbe spherical distance between
eiandej);
(ii) ifeiandejare unit outer normals of two unit outer normal of oriented,then gij=coseiej”(whereeiej”is dihedral angle formed byeiandej);
(iii) if either ofeiandejis a point,and another is an outer normal,thengij=sinhij
(wherehijis the spherical distance with sign based on the direction from the point to
the hyperplane).
If m>n+ ,then
det(gij)mi,j== .
Lemma Let hibe the altitude from vertex Piof an n-simplexnin Sn()for i= , , . . . , n+ ,and M= (cosρij)n+i,j=the edge matrix,then we have
sinhi=
|M|
Mii (i= , , . . . ,n+ ). ()
Proof of Theorem Leteiis the unit outer normal of the orientedfi(i= , , . . . ,n+ ) and the pointen+=D, such that”eiej=π–ϕij(i,j= , , . . . ,n+) and the spherical distance with sign based on the direction from the pointen+to the hyperplaneeiisrifori= , , . . . ,n+ .
By Lemma we know that the (n+ )×(n+ )-order matrix (coseiej)”n+
i,j== (–cosϕij)n+i,j== Ais a positive definite symmetric matrix. Becauseλi= (i= , , . . . ,n+ ), the matrix T= (–λiλjcosϕij)n+i,j=is also a positive definite symmetric matrix.
By Lemma we have
B=
sinr
–cosϕij ...
sinrn+
sinr · · · sinrn+
= . ()
From () andλi= (i= , , . . . ,n+ ), we get
detG=
λsinr
–λiλjcosϕij ...
λn+sinrn+
λsinr · · · λn+sinrn+
= . ()
Because the matrixT= (–λiλjcosϕij)n+i,j=is also a positive definite symmetric matrix and
detG= , the matrixGis a semi-positive definite symmetric matrix and the rank of matrix Gisn+ . Letui> (i= , , . . . ,n+ ) andun+= be the eigenvalues of the matrixG, and
σ= n+
i= ui=
n+
i=
ui, σ=
≤i<j≤n+
uiuj=
≤i<j≤n+ uiuj.
Using Maclaurin’s inequality [], we have
Å
n+ σ
ã
≥
n(n+ )σ. ()
Equality holds if and only ifu=u=· · ·=un+.
By the relation between the eigenvalues and the principal minors of the matrixG, we have
σ= n+
i=
λi+ , σ=
≤i<j≤n+
λiλj sinϕij+ n+
i=
λicosri. ()
Substituting () into (), we get inequality (). It is easy to see that equality holds in () if and only if the nonzero eigenvalues of matrixGare all equal.
and by Lemma we have
cosR
cosρij ...
cosR
cosR · · · cosR
= .
From this andxi> (i= , , . . . ,n+ ), we get
detB=
√
xcosR
√
xixjcosρij ...
√
xn+cosR
√
xcosR · · · √xn+cosR
= . ()
Because the matrixQ= (√xixjcosρij)n+i,j= is positive definite symmetric anddetB= , the matrixBis a semi-positive definite symmetric matrix and its rank isn+ . Letvi> (i= , , . . . ,n+ ),vn+= be the eigenvalues of matrixB, and
σ= n+
i= vi=
n+
i=
vi, σ=
≤i<j≤n+
vivj=
≤i<j≤n+ vivj.
Using Maclaurin’s inequality [], we have
Å
n+ σ
ã
≥
n(n+ )σ. ()
Equality holds if and only ifv=v=· · ·=vn+.
By the relation between the eigenvalues and the principal minors of the matrixB, we have
σ= n+
i=
xi+ , σ=
≤i<j≤n+
xixjsinρij+ n+
i=
xi –cosR. ()
Substituting () into (), we get inequality (). It is easy to see that equality holds in () if and only if the nonzero eigenvalues of matrixBare all equal.
Proof of Theorem Fromxi> (i= , , . . . ,n+ ) and the edge matrixM= (cosρij)n+i,j= ofnbeing a positive definite symmetric matrix, we know that the matrixQin () is also a positive definite symmetric matrix. Letai> (i= , , . . . ,n+ ) be the eigenvalues of the matrixQ, and
σn= n+
i= n+
j= j=i
aj, σn+= n+
By Maclaurin’s inequality [], we have
Å
n+ σn
ã n
≥(σn+)n+ . ()
Equality holds if and only ifa=a=· · ·=an+.
By the relation between the eigenvalues and the principal minors of the matrixQ, we have
σn= n+
i= Qii=
n+
i=
Çn+
j= j=i
xj
å
Mii (i= , , . . . ,n+ ), ()
σn+=|Q|=
Çn+
i= xi
å
· |M|. ()
From (), (), and (), we get
n+
i=
Çn+
j= j=i
xj
å
Mii≥(n+ )
Çn+
i= xi
å n n+
· |M|n+n . ()
By Lemma we have
Mii=|M|cschi (i= , , . . . ,n+ ). () Substituting () into (), we get inequality (). It is easy to see that equality holds in () if and only if the eigenvalues of matrixQare all equal.
Finally, we prove that inequality () is valid:
|M|≤
ï
n(n+ )
≤i<j≤n+
sinρij
òn+
. ()
Letbi(i= , , . . . ,n+ ) be the eigenvalues of the edge matrixM= (cosρij)n+i,j=. Since the matrixMis a positive definite symmetric matrix,bi> . Let
σ=
≤i<j≤n+
bibj, σn+=
n+
i= bi.
By Maclaurin’s inequality [], we have
Å
n(n+ )σ
ã
≥(σn+)
n+. ()
Equality holds if and only ifb=b=· · ·=bn+.
By the relation between the eigenvalues and the principal minors of the matrixM, we have
σ=
≤i<j≤n+
From () and (), we get inequality (). Ifnis a regular simplex inSn(), thenρij=π (i=j,i,j= , , . . . ,n+ ),|M|= andMii= (i= , , . . . ,n+ ). By () we havesinhi= (i= , , . . . ,n+ ); thus equality holds in () ifnis a regular simplex.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors co-authored this paper. All authors read and approved the final manuscript.
Author details
1Department of Mathematics and Teachers Educational Research, Hefei Normal University, Hefei, 230601, P.R. China. 2Anhui Xinhua University, Hefei, 230088, P.R. China.3School of mathematical Since, Anhui University, Hefei, 230039, P.R.
China.
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.
Received: 26 August 2013 Accepted: 20 January 2014 Published:10 Feb 2014 References
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Cite this article as:Shi-guo et al.:Inequalities for ann-simplex in spherical spaceSn(1).Journal of Inequalities and