Journal of Inequalities in Pure and
Applied Mathematics
THE GENERALIZED SINE LAW AND SOME INEQUALITIES
FOR SIMPLICES
volume 5, issue 4, article 106,
2004.
SHIGUO YANG
Department of Mathematics
Anhui Institute of Education
Hefei 230061
People’s Republic of China.
Received 31 July, 2003;
accepted 23 April, 2004.
Communicated by: A. Sofo
EMail: sxx@ahieedu.net.cn
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
106-03
Quit
Abstract
The sines of k-dimensional vertex angles of an n-simplex is defined and the
law of sines for k-dimensional vertex angles of an n-simplex is established.
Using the generalized sine law for n-simplex, we obtain some inequalities for
the sines of k-dimensional vertex angles of an n-simplex. Besides, we obtain
inequalities for volumes of n-simplices. As corollaries, the generalizations to
several dimensions of the Neuberg-Pedoe inequality and P. Chiakuei inequality,
and an inequality for pedal simplex are given.
The Generalized Sine Law and
Some Inequalities for Simplices
Shiguo Yang
2000 Mathematics Subject Classification: 52A40.
Key words: Simplex, k-dimensional vertex angle, Volume, Inequality.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
The Generalized Sine Law for Simplices . . . . . . . . . . . . . . . . . 4
3
Some Inequalities for Volumes of Simplices . . . . . . . . . . . . . . 11
References
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 23
J. Ineq. Pure and Appl. Math. 5(4) Art. 106, 2004
http://jipam.vu.edu.au
1.
Introduction
The law of sines for triangles in E 2 has natural analogues in higher dimensions.
In 1978, F. Eriksson [1] defined the n-dimensional sines of the n-dimensional
corners of an n-simplex in n-dimensional Euclidean space E n and obtained the
law of sines for the n-dimensional corners of an n-simplex. In this paper, the
sines of k-dimensional vertex angles of an n -simplex will be defined, and the
law of sines for k-dimensional vertex angles of an n-simplex will be established.
Using the generalized sine law for simplices and a known inequality in [2],
we get some inequalities for the sines of k-dimensional vertex angles of an nsimplex.
Recently, Yang Lu and Zhang Jingzhong [2, 3], Yang Shiguo [4], Leng
Gangson [5, 6] and D. Veljan [7] and V. Volenec et al. [9] have obtained
some important inequalities for volumes of n-simplices. In this paper, some
interesting new inequalities for volumes of n-simplices will be established. As
corollaries, we will obtain an inequality for pedal simplex and a generalization
to several dimensions of the Neuberg-Pedoe inequality, which differs from the
results in [4], [5] and [6].
The Generalized Sine Law and
Some Inequalities for Simplices
Shiguo Yang
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 3 of 23
J. Ineq. Pure and Appl. Math. 5(4) Art. 106, 2004
http://jipam.vu.edu.au
2.
The Generalized Sine Law for Simplices
Let Ai (i = 1, 2, . . . , n + 1) be the vertices of an n-dimensional simplex Ωn in
the n-dimensional Euclidean space E n , V the volume of the simplex Ωn and
Fi (n − 1)-dimensional content of Ωn . F. Eriksson defined the n-dimensional
sines of the n-dimensional corners αi of the n-simplex Ωn and obtained the law
of sines for n-simplices as follows [1]
(2.1)
Fi
=
n sin α
i
(n − 1)!
n+1
Q
j=1
(nV )n−1
Fj
(i = 1, 2, . . . , n + 1).
Shiguo Yang
In this paper, we will define the sines of the k-dimensional vertex angles of
an n-dimensional simplex and establish the law of sines for the k-dimensional
vertex angles of an n-simplex. Let Vi1 i2 ···ik be the (k − 1)-dimensional content
of the (k − 1)-dimensional face Ai1 Ai2 · · · Aik ((k − 1)-simplex) of the simplex
Ωn for k ∈ {2, 3, . . . , n + 1} and i1 , i2 , . . . , ik ∈ {1, 2, . . . , n + 1}, O and
R denote the circumcenter and circumradius of the simplex Ωn respectively.
−→
OAi = Rui (i = 1, 2, . . . , n + 1), ui is the unit vector. The sines of the kdimensional vertex angles of the simplex Ωn are defined as follows.
Definition 2.1. Let αij denote the angle formed by the vectors ui and uj . The
sine of a k-dimensional vertex angle ϕi1 i2 ···ik of the simplex Ωn corresponding
the (k − 1)-dimensional face Ai1 Ai2 · · · Aik is defined as
(2.2)
The Generalized Sine Law and
Some Inequalities for Simplices
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 4 of 23
1
sin ϕi1 i2 ···ik = (−Di1 i2 ···ik ) 2 ,
J. Ineq. Pure and Appl. Math. 5(4) Art. 106, 2004
http://jipam.vu.edu.au
where
(2.3)
Di1 i2 ···ik
=
0 1
···
1
..
α
.
− 12 sin2 il2im
1
1 (l, m = 1, 2, . . . , k).
We will prove that
1
0 < (−Di1 i2 ···ik ) 2 ≤ 1.
(2.4)
If n = 2, the sine of the 2-dimensional vertex angle ϕij of the triangle A1 A2 A3
is the sine of the angle formed by two edges Ak Ai and Ak Aj .
With the notation introduced above, we establish the law of sines for the k
-dimensional vertex angles of an n-simplex as follows.
Theorem 2.1. For an n-dimensional simplex Ωn in E n and k ∈ {2, 3, . . . , n +
1}, we have
(2.5)
Vi1 i2 ···ik
(2R)k−1
=
sin ϕi1 i2 ···ik
(k − 1)!
(1 ≤ i1 < i2 < · · · < ik ≤ n + 1).
Put ϕ12···i−1,i+1,...,n+1 = θi , V12···i−1,i+1,...,n+1 = Fi (i = 1, 2, . . . , n + 1), by
Theorem 2.1 we obtain the law of sines for the n-dimensional vertex angles of
n-simplices as follows.
Corollary 2.2.
(2.6)
The Generalized Sine Law and
Some Inequalities for Simplices
Shiguo Yang
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 5 of 23
F1
F2
Fn+1
(2R)n−1
=
= ··· =
=
.
sin θ1
sin θ2
sin θn+1
(n − 1)!
J. Ineq. Pure and Appl. Math. 5(4) Art. 106, 2004
http://jipam.vu.edu.au
If we take n = 2 in Theorem 2.1 or Corollary 2.2, we obtain the law of sines
for a triangle A1 A2 A3 in the form
a1
a2
a3
(2.7)
=
=
= 2R.
sin A1
sin A2
sin A3
Proof of Theorem 2.1. Let aij = |Ai Aj | (i, j = 1, 2, . . . , n + 1), then
αij
aij = 2R sin
,
2
The Generalized Sine Law and
Some Inequalities for Simplices
(2.8) sin2 ϕi1 i2 ···ik
= −
0 1
···
1 1
= −Di1 i2 ···ik
..
1
2
.
− 8R2 ail im
1
0 1 ··· 1 1
= (−1)k (8R2 )−(k−1) · ..
2
.
a
i
i
m
l
1
Shiguo Yang
Title Page
Contents
(l, m = 1, 2, . . . , k).
By the formula for the volume of a simplex, we have
(2.9)
sin2 ϕi1 i2 ···ik = −Di1 i2 ···ik
= (−1)k (8R2 )−(k−1) (−1)k 2k−1 (k − 1)!2 Vi21 i2 ···ik
(k − 1)!2 2
=
V
.
(2R)2(k−1) i1 i2 ···ik
JJ
J
II
I
Go Back
Close
Quit
Page 6 of 23
J. Ineq. Pure and Appl. Math. 5(4) Art. 106, 2004
http://jipam.vu.edu.au
From this equality (2.5) follows.
Now we prove that inequality (2.4) holds. When k ≥ 2, we have k ≤ 2k−1 .
Using the Voljan-Korchmaros inequality [3], we have
(2.10)
Vi1 i2 ···ik
1
≤
(k − 1)!
12
k
2k−1
! k2
Y
ail im
.
1≤l<m≤k
Equality holds if and only if the simplex Ai1 Ai2 · · · Aik is regular.
Combining inequality (2.10) with equality (2.5), we get
(2.11)
Vi1 i2 ···ik
(2R)k−1
≤
·
(k − 1)!
(2R)k−1
≤
·
(k − 1)!
(2R)k−1
≤
.
(k − 1)!
k
12
2k−1
k
αi i
sin l m
2
1≤l<m≤k
! k2
Title Page
Contents
2k−1
JJ
J
Using equality (2.5) and inequality (2.11), we get
1
Shiguo Yang
Y
21
0 < (−Di1 i2 ···ik ) 2 = sin ϕi1 i2 ···ik =
The Generalized Sine Law and
Some Inequalities for Simplices
(k − 1)!
Vi i ···i ≤ 1.
(2R)k−1 1 2 k
II
I
Go Back
Close
Quit
Page 7 of 23
For the sines of the k-dimensional vertex angles of an n-simplex, we obtain
an inequality as follows.
J. Ineq. Pure and Appl. Math. 5(4) Art. 106, 2004
http://jipam.vu.edu.au
Theorem 2.3. Let ϕi1 i2 ···ik (1 ≤ i1 < i2 < · · · < ik ≤ n + 1) denote
the k-dimensional vertex angles of an n-simplex Ωn in E n , and λi > 0 (i =
1, 2, . . . , n + 1) be arbitrary real numbers, then we have
X
(2.12)
λi1 λi2 · · · λik sin2 ϕi1 i2 ···ik
1≤i1 <i2 <···<ik ≤n+1
Pn+1 k
n! ·
i=1 λi
≤
.
(n − k + 1)!(k − 1)!(4n)k−1
Equality holds if λ1 = λ2 = · · · = λn+1 and the simplex Ωn is regular.
Shiguo Yang
By taking λ1 = λ2 = · · · = λn+1 in the inequality (2.12), we get:
Corollary 2.4.
Title Page
X
(2.13)
sin2 ϕi1 i2 ···ik ≤
1≤i1 <i2 <···<ik ≤n+1
n! · (n + 1)k
.
(n − k + 1)!(k − 1)!(4n)k−1
Equality holds if the simplex Ωn is regular.
Lemma 2.5. Let Ωn be an n-simplex in E n , xi > 0 (i = 1, 2, . . . , n + 1) be real
numbers, Vi1 i2 ···is+1 be the s-dimensional volume of the s-dimensional simplex
Ai1 Ai2 · · · Ais+1 for i1 , i2 , . . . , is+1 ∈ {1, 2, . . . , n + 1}. Put
X
1≤i1 <i2 <···<ik ≤n+1
xi1 xi2 · · · xis+1 Vi21 i2 ···is+1 , M0 =
Contents
JJ
J
II
I
Go Back
To prove Theorem 2.3, we need a lemma as follows.
Ms =
The Generalized Sine Law and
Some Inequalities for Simplices
n+1
X
i=1
xi ,
Close
Quit
Page 8 of 23
J. Ineq. Pure and Appl. Math. 5(4) Art. 106, 2004
http://jipam.vu.edu.au
then we have
(2.14)
Msl ≥
[(n − l)!(l!)3 ]s
(n! · M0 )l−s Mls (1 ≤ s < l ≤ n).
[(n − s)!(s!)3 ]l
Equality holds if and only if the intertial ellipsoid of the points A1 , A2 , . . . , An+1
with masses x1 , x2 , . . . , xn+1 is a sphere.
For the proof of Lemma 2.5. the reader is referred to [2] or [9].
Proof of Theorem 2.3. By putting s = 1, l = k−1 and xi = λi (i = 1, 2, . . . , n+
1) in the inequality (2.14), we have
!k−1
X
(2.15)
λi λj a2ij
The Generalized Sine Law and
Some Inequalities for Simplices
Shiguo Yang
1≤i<j≤n+1
≥
(n − k + 1)! · (k − 1)!
[(n − 1)!]k−1
3
n! ·
n+1
X
!k−2
λi
i=1
X
×
λi1 λi2 · · · λik Vi21 i2 ···ik .
1≤i1 <i2 <···<ik ≤n+1
By Theorem 2.1, we have
(2.16)
Vi1 i2 ···ik =
X
1≤i<j≤n+1
Contents
JJ
J
II
I
Go Back
(2R)k−1
sin ϕi1 i2 ···ik .
(k − 1)!
Using the known inequality [3]
(2.17)
Title Page
Close
Quit
Page 9 of 23
λi λj a2ij ≤
n+1
X
i=1
!2
λi
R2 ,
J. Ineq. Pure and Appl. Math. 5(4) Art. 106, 2004
http://jipam.vu.edu.au
P
with equality if and only if the point P = n+1
i=1 λi Ai is the circumcenter of
simplex Ωn .
Combining (2.15) with (2.16) and (2.17), we obtain inequality (2.12). It is
easy to see that equality holds in (2.12) if λ1 = λ2 = · · · = λn+1 and simplex
Ωn is regular.
The Generalized Sine Law and
Some Inequalities for Simplices
Shiguo Yang
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 10 of 23
J. Ineq. Pure and Appl. Math. 5(4) Art. 106, 2004
http://jipam.vu.edu.au
3.
Some Inequalities for Volumes of Simplices
Let P be an arbitrary point inside the simplex Ωn and Bi the orthogonal projection of the point P on the (n − 1)-dimensional plane σi containing (n −
1)-simplex fi = A1 · · · Ai−1 Ai+1 · · · An+1 . Simplex Ωn = B1 B2 · · · Bn+1 is
called the pedal simplex of the point P with respect to the simplex Ωn . Let
ri = |P Bi | (i = 1, 2, . . . , n + 1), V be the volume of the pedal simplex
Ωn , V (i) and V (i) denote the volumes of two n-dimensional simplices Ωn (i) =
A1 · · · Ai−1 P Ai+1 · · · An+1 and Ωn (i) = B1 · · · Bi−1 P Bi+1 · · · Bn+1 , respectively. Then we have an inequality for volumes of just defined n-simplices as
follows.
Theorem 3.1. Let P be an arbitrary point inside n-dimensional simplex Ωn and
λi (i = 1, 2, . . . , n + 1) positive real numbers, then we have
" n
#n
n+1
X
1 X
λ1 · · · λi−1 λi+1 · · · λn+1 V (i) ≤ n
λi V (i) V 1−n ,
(3.1)
n i=1
i=1
with equality if the simplex Ωn is regular, P is the circumcenter of Ωn and
λ1 = λ2 = · · · = λn+1 .
Now we state some applications of Theorem 3.1.
If taking λ1 = λ2 = · · · = λn+1 in inequality (3.1), we have
" n+1
#n
n+1
X
1 X
(3.2)
V (i) ≤ n
V (i) · V 1−n .
n
i=1
i=1
The Generalized Sine Law and
Some Inequalities for Simplices
Shiguo Yang
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 11 of 23
J. Ineq. Pure and Appl. Math. 5(4) Art. 106, 2004
http://jipam.vu.edu.au
Since the point P is in the interior of the simplex Ωn , then
n+1
X
(3.3)
n+1
X
V (i) = V ,
i=1
V (i) = V.
i=1
Using (3.2) and (3.3) we obtain an inequality for the volume of the pedal simplex Ωn of the point P with respect to the simplex Ωn as follows.
Corollary 3.2. Let P be an arbitrary point inside the n-simplex Ωn , then we
have
V ≤
(3.4)
1
V,
nn
The Generalized Sine Law and
Some Inequalities for Simplices
Shiguo Yang
with equality if simplex Ωn is regular and P is the circumcenter of Ωn .
Title Page
Corollary 3.3. Let P be an arbitrary point inside the n-simplex Ωn , then we
have
Contents
n+1
X
(3.5)
V (i) · V (i) ≤
i=1
1
V 2,
n
(n + 1)n
JJ
J
II
I
Go Back
with equality if the simplex Ωn is regular and P is the circumcenter of Ωn .
Proof. Let λi = [V (i)]−1 (i = 1, 2, . . . , n + 1) in inequality (3.1); we get
Close
Quit
Page 12 of 23
(3.6)
n+1
X
i=1
V (i) · V (i) ≤
n+1
n
n
V 1−n
n+1
Y
j=1
V (j).
J. Ineq. Pure and Appl. Math. 5(4) Art. 106, 2004
http://jipam.vu.edu.au
Using the arithmetic-geometric mean inequality and equality (3.3), we have
"
#n+1
n
n+1
n+1
X
X
n+1
1
1
V (i) · V (i) ≤
V 1−n
V (j)
=
V 2.
n
n
n
+
1
(n
+
1)n
i=1
j=1
It is easy to see that equality in (3.5) holds if the simplex Ωn is regular and the
point P is the circumcenter of Ωn .
−−→
Proof of Theorem 3.1. Let hi be the altitude of simplex Ωn from vertex Ai , P B i
= ri ei , where ei is the unit outer normal vector of the i-th side face fi = A1 · · ·
Ai−1 Ai+1 · · · An+1 of the simplex Ωn , and n sin αk be the n-dimensional sine of
the k-th corner αk of the simplex Ωn . Wang and Yang [8] proved that
n
(3.7)
1
sin αn = [det(ei · ej )ij6=k ] 2
Shiguo Yang
(k = 1, 2, . . . , n + 1).
By the formula for the volume of an n-simplex and (3.7), we have


n+1
1
1
1 Y  n
(3.8)
V (i) =
[det(rl rk el · ek )l,k6=i ] 2 =
rj  · sin αi .

n!
n!
j=1
j6=i
Using (3.8), (2.1) and nV = hi Fi , we get that
n+1
X
The Generalized Sine Law and
Some Inequalities for Simplices
Title Page
Contents
JJ
J
II
I
Go Back
Close
λ1 · · · λi−1 λi+1 · · · λn+1 V (i)
Quit
i=1
=
1
n!
n+1
X
i=1



n+1
Y
j=1
j6=i


λj rj  · n sin αi
Page 13 of 23
J. Ineq. Pure and Appl. Math. 5(4) Art. 106, 2004
http://jipam.vu.edu.au

=
1
n!
n+1 

X
n+1
Y
i=1
j=1
j6=i







λj rj  (nV )n−1 (n − 1)! ·
= [(n!) · V ]
−1
n+1
Y
j=1
j6=i

2

−1 



Fj 



n+1
n+1
X
Y

λj rj hj  ,

i=1
j=1
j6=i
The Generalized Sine Law and
Some Inequalities for Simplices
i.e.
(3.9)
2
(n!) V
n+1
X
λ1 · · · λi−1 λi+1 · · · λn+1 V (i) =
i=1
n+1
X
i=1



n+1
Y

j=1
j6=i
Taking s = n − 1, l = n in inequality (2.14), we get


 n
! n+1 !n−1
n+1
n+1
n+1
3n
X
Y
X
Y
n


 
(3.10)
xj  Fi2  ≥
x
xi
V 2(n−1) .


i
2
(n!)
i=1
i=1
i=1
j=1
j6=i
Let xi = (λi ri hi )−1 (i = 1, 2, . . . , n + 1) in inequality (3.10). Then we have


 
!n
n+1
n+1
n+1
X
n3n X Y
 
(3.11)
λi ri hi Fi2
≥
λj rj hj  Fi2  · V 2(n−1) .


2
(n!) i=1
i=1
j=1
j6=i
Shiguo Yang

λj rj hj  .
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 14 of 23
J. Ineq. Pure and Appl. Math. 5(4) Art. 106, 2004
http://jipam.vu.edu.au
Using inequality (3.11) and ri Fi = nV (i), hi Fi = nV , we get


" n+1
#n
n+1
n+1
X
nn 2(n−1) X Y

λj rj hj  .
(3.12)
Vn
V
λi V (i) ≥

2
(n!)
i=1
i=1
j=1
j6=i
Substituting equality (3.9) into inequality (3.12) we get inequality (3.1). It is
easy to prove that equality in (3.1) holds if simplex Ωn is regular, P is the
circumcenter of Ωn and λ1 = λ2 = · · · = λn+1 . Theorem 3.1 is proved.
Finally, we shall establish some inequalities for volumes of two n-simplices.
As corollaries, the generalizations to several dimensions of the Neuberg-Pedoe
inequality and P.Chiakui inequality will be given.
Let ai (i = 1, 2, 3) denote the sides of the triangle A1 A2 A3 with area ∆, and
0
ai (i = 1, 2, 3) denote the sides of the triangle A01 A02 A03 with area ∆0 , then
!
3
3
X
X
2
2
(3.13)
a2i
a0j − 2 (a0i ) ≥ 16∆∆0 ,
i=1
j=1
with equality if and only if ∆A1 A2 A3 is similar to ∆A01 A02 A03 .
Inequality (3.13) is the well-known Neuberg-Pedoe inequality.
In 1984, P. Chiakui [9] proved the following sharpening of the Neuberg-
The Generalized Sine Law and
Some Inequalities for Simplices
Shiguo Yang
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 15 of 23
J. Ineq. Pure and Appl. Math. 5(4) Art. 106, 2004
http://jipam.vu.edu.au
Pedoe inequality:
(3.14)
3
X
a2i
3
X
i=1
≥8
!
0 2
aj
−
2
2 (a0i )
j=1
(a01 )2 + (a02 )2 + (a03 )2 2
a21 + a22 + a23
2
∆
+
(∆0 )
0 2
0 2
0 2
a21 + a22 + a23
(a1 ) + (a2 ) + (a3 )
!
,
with equality if and only if ∆A1 A2 A3 is similar to ∆A01 A02 A03 .
Recently, Leng Gangson [5] has extended inequality (3.14) to the edge lengths
and volumes of two n-simplices. In this paper, we shall extend inequality (3.14)
to the volumes of two n-simplices and the contents of their side faces. As a
corollary, we get a generalization to several dimensions of the Neuberg-Pedoe
inequality. Let Ai (i = 1, 2, . . . , n + 1) be the vertices of n-simplex Ωn in
E n , V the volume of the simplex Ωn and Fi (n − 1)- dimensional content of
the (n − 1)-dimensional face fi = A1 · · · Ai−1 Ai+1 · · · An+1 of Ωn . For two
n-simplices Ωn and Ω0n and real numbers α, β ∈ (0, 1], we put
(3.15)
σn (α) =
n+1
X
i=1
Fiα ,
σn (β) =
n+1
X
i=1
β
(Fi0 )
,
n3
bn =
n+1
n+1
n!2
n1
.
We obtain an inequality for volumes of two n-simplices as follows.
Theorem 3.4. For any two n-dimensional simplices Ωn and Ω0n and two arbi-
The Generalized Sine Law and
Some Inequalities for Simplices
Shiguo Yang
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 16 of 23
J. Ineq. Pure and Appl. Math. 5(4) Art. 106, 2004
http://jipam.vu.edu.au
trary real numbers α, β ∈ (0, 1], we have
!
n+1
n+1
X
X
β
β
Fj0 − 2 (Fi0 )
(3.16)
Fiα
i=1
j=1
(n − 1)2 α σn (β) 2(n−1)α/n
β σn (α)
0 2(n−1)β/n
≥
bn
V
+ bn
(V )
.
2
σn (α)
σn (β)
Equality holds if and only if simplices Ωn and Ω0n are regular.
Using inequality (3.16) and the arithmetic-geometric mean inequality, we
get the following corollary.
The Generalized Sine Law and
Some Inequalities for Simplices
Shiguo Yang
Ω0n
Corollary 3.5. For any two n-dimensional simplices Ωn and
and two arbitrary real numbers α, β ∈ (0, 1], we have
!
n+1
n+1
X
X
β
β
β
≥ b(α+β)/2
(n2 − 1)(V α (V 0 ) )(n−1)/n .
(3.17)
Fiα
Fj0 − 2 (Fi0 )
n
i=1
j=1
Equality holds if and only if simplices Ωn and Ω0n are regular.
If we let α = β in Corollary 3.5, we get Leng Gangson’s inequality [6] as
follows. For any θ ∈ (0, 1] we have
!
n+1
n+1
X
X
θ
0 θ
0 θ
(3.18)
Fi
Fj − 2 (Fi ) ≥ bθn (n2 − 1)(V V 0 )(n−1)θ/n ,
i=1
j=1
with equality if and only if simplices Ωn and Ω0n are regular.
To prove Theorem 3.4, we need some lemmas as follows.
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 17 of 23
J. Ineq. Pure and Appl. Math. 5(4) Art. 106, 2004
http://jipam.vu.edu.au
Lemma 3.6. For an n-simplex Ωn and arbitrary number α ∈ (0, 1], we have
Qn+1 2α
3n α
1
n
i=1 Fi
V 2(n−1)α ,
(3.19)
Pn+1 2α ≥
(n−1)α+1
2
(n + 1)
(n!)
i=1 Fi
with equality if and only if simplex Ωn is regular.
Proof. If taking l = n, s = n−1 and xi = Fi2 (i = 1, 2, . . . , n+1) in inequality
(2.14), we get an inequality as follows
2 n+1
Y
n
(n + 1) (n!)
n3n
Fi2 ≥ V 2(n−1)
i=1
n+1
X
The Generalized Sine Law and
Some Inequalities for Simplices
Fi2 ,
Shiguo Yang
i=1
or
(3.20)
Title Page
2α n+1
Y
(n + 1)nα (n!)
n3nα
Fi2α ≥ V 2(n−1)α
i=1
n+1
X
!α
Fi2
.
i=1
It is easy to prove that equality in (3.20) holds if and only if simplex Ωn is
regular. From inequality (3.20) we know that inequality (3.19) holds for α = 1.
For α ∈ (0, 1), using inequality (3.20) and the well-known inequality
Contents
JJ
J
II
I
Go Back
Close
(3.21)
n+1
X
i=1
Fi2 ≥ (n + 1)
1
n+1
n+1
X
! α1
Fi2α
,
i=1
we get inequality (3.19). It is easy to see that equality in (3.19) holds if and only
if the simplex Ωn is regular.
Quit
Page 18 of 23
J. Ineq. Pure and Appl. Math. 5(4) Art. 106, 2004
http://jipam.vu.edu.au
Lemma 3.7. For an n-simplex Ωn (n ≥ 3) and an arbitrary number α ∈ (0, 1],
we have
!2
n+1
n+1
X
X
α
(3.22)
Fi
−2
Fi2α ≥ bαn (n2 − 1)V 2(n−1)α/n ,
i=1
i=1
with equality if and only if the simplex Ωn is regular.
For the proof of Lemma 3.7, the reader is referred to [6].
Lemma 3.8. Let ai (i = 1, 2, 3) and ∆ denote the sides and the area of the
triangle (A1 A2 A3 ), respectively. For arbitrary number α ∈ (0, 1], denote by
∆α the area of the triangle (A1 A2 A3 )α with sides aαi (i = 1, 2, 3), then the
following inequality holds
α
3 16 2
2
(3.23)
∆α ≥
∆
.
16 3
For α 6= 1, equality holds if and only if a1 = a2 = a3 .
(3.24)
σn σn0 − 2
n+1
X
Pn+1
i=1
Title Page
Contents
xi ,
II
I
Go Back
Close
Quit
xi yi
Page 19 of 23
i=1
"
≥
Shiguo Yang
JJ
J
For the proof of Lemma 3.8, the reader is referred to [9].
Lemma
P 3.9. Let numbers xi > 0, yi > 0 (i = 1, 2, . . . , n + 1), σn =
σn0 = n+1
i=1 yi , then
The Generalized Sine Law and
Some Inequalities for Simplices
σn0
1
2 σn
σn2 − 2
n+1
X
i=1
!
x2i
+
σn
σn0
2
(σn0 ) − 2
n+1
X
i=1
!#
yi2
,
J. Ineq. Pure and Appl. Math. 5(4) Art. 106, 2004
http://jipam.vu.edu.au
with equality if and only if
y1
y2
yn+1
=
= ··· =
.
x1
x2
xn+1
Proof. Inequality (3.24) is
(3.25)
n+1
n+1
n+1
X
σn0 X 2 σn X 2
x +
y ≥2
xi yi .
σn i=1 i σn0 i=1 i
i=1
Now we prove that inequality (3.25) holds. Using the arithmetic-geometric
mean inequality, we have
σn0 2 σn 2
x + y ≥ 2xi yi
σn i σn0 i
The Generalized Sine Law and
Some Inequalities for Simplices
Shiguo Yang
(i = 1, 2, . . . , n + 1).
Adding up those n +0 1 inequalities, we get inequality (3.25). Equality in (3.25)
holds if and only if σσnn x2i = σσn0 yi2 (i = 1, 2, . . . , n + 1), i.e.
n
y1
y2
yn+1
σ0
=
= ··· =
= n.
x1
x2
xn+1
σn
Title Page
Contents
JJ
J
II
I
Go Back
Close
Proof of Theorem 3.4. For n = 2, consider two triangles (A1 A2 A3 )α and (A01 A02 A03 )β .
Using inequality (3.14) and Lemma 3.8, we have
!
3
3
X
X
1 α σ20 (β) α
β σ2 (α)
0 β
α
0 β
0 β
(3.26)
b2
∆ + b2 0
(∆ ) .
ai
aj − 2 (ai )
≥
2
σ
(α)
σ
(β)
2
2
i=1
j=1
Quit
Page 20 of 23
J. Ineq. Pure and Appl. Math. 5(4) Art. 106, 2004
http://jipam.vu.edu.au
Equality in (3.26) holds if and only if a1 = a2 = a3 and a01 = a02 = a03 . Hence,
inequality (3.16) holds for n = 2.
For n ≥ 3, taking xi = Fiα , yi = (Fi0 )β (i = 1, 2, . . . , n + 1) in inequality
(3.24), we get
!
n+1
n+1
X
X
β
β
(3.27)
Fiα
Fj0 − 2 (Fi0 )
i=1
j=1
=
n+1
X
!
Fiα
n+1
X
!
β
(Fi0 )
−2
n+1
X
β
Fiα (Fi0 )
i=1
i=1

 i=1

!
2
n+1
n+1

0
X
X
1 σn (β) 
≥
Fiα − 2
Fi2α 
2  σn (α)
i=1
i=1


!2
n+1
n+1
X 2β 
σn (α)  X 0 β
+ 0
(Fi )
−2
Fi  .

σn (β)
i=1
i=1
Using inequality (3.27) and Lemma 3.7, we get
!
n+1
n+1
X
X
n2 − 1 α σn0 (β) 2(n−1)α/n
α
0 β
β σn (α) 2(n−1)β/n
bn
V
+ bn 0
V
Fi
Fj
≥
.
2
σ
(α)
σ
(β)
n
n
i=1
i=1
Hence, inequality (3.16) is true for n ≥ 3. For n ≥ 3, it is easy to see that
equality in (3.16) holds if and only if two simplices Ωn and σn0 are regular.
Theorem 3.4 is proved.
The Generalized Sine Law and
Some Inequalities for Simplices
Shiguo Yang
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 21 of 23
J. Ineq. Pure and Appl. Math. 5(4) Art. 106, 2004
http://jipam.vu.edu.au
References
[1] F. ERIKSSON, The law of sines for tetrahedra and n-simplics, Geometriae
Dedicata, 7 (1978), 71–80.
[2] JINGZHONG ZHANG AND LU YANG, A class of geometric inequalities concerning the mass-points system, J. China. Univ. Sci. Techol., 11(2)
(1981), 1–8.
[3] JINGZHONG ZHANG AND LU YANG, A generalization to several dimensions of the Neuberg-Pedoe inequality, Bull. Austral. Math. Soc., 27
(1983), 203–214.
[4] SHIGUO YANG, Three geometric inequalities for a simplex, Geometriae
Dedicata, 57 (1995), 105–110.
[5] GANGSON LENG, Inequalities for edge-lengths and volumes of two simplices, Geometriae Dedicata, 68 (1997), 43–48.
[6] GANGSON LENG, Some generalizations to several dimensions of the Pedoe inequality with applications, Acta Mathematica Sinica, 40 (1997), 14–
21.
The Generalized Sine Law and
Some Inequalities for Simplices
Shiguo Yang
Title Page
Contents
JJ
J
II
I
Go Back
[7] D. VELJAN, The sine theorem and inequalities for volumes of simplices
and determinants, Lin. Alg. Appl., 219 (1995), 79–91.
Close
[8] SHIGUO YANG AND JIA WANG, An inequality for n-dimensional sines
of vertex angles of a simplex with some aplications, Journal of Geometry,
54 (1995), 198–202.
Page 22 of 23
Quit
J. Ineq. Pure and Appl. Math. 5(4) Art. 106, 2004
http://jipam.vu.edu.au
[9] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND V. VOLENEC, Recent Geometric Inequalities, Kluwer, Dordrecht, Boston, London, 1989.
The Generalized Sine Law and
Some Inequalities for Simplices
Shiguo Yang
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 23 of 23
J. Ineq. Pure and Appl. Math. 5(4) Art. 106, 2004
http://jipam.vu.edu.au