J. geom. 75 (2002) 123 – 131
0047–2468/02/020123 – 09
© Birkhäuser Verlag, Basel, 2002
DOI 10.1007/s00022-002-1527-4
Triangle inequalities associated with the vertex angles
and dihedral angles of a simplex and their applications
Sun Mingbao∗
Abstract. In this paper, a class of triangle inequalities associated with the vertex angles and dihedral angles of a
simplex is given. As an application, we improve an inequality on the volume of the pedal simplex.
Mathematics Subject Classification (2000): 52A40, 51K05.
Key words: Simplex, vertex angle, dihedral angle, volume, inequality.
1. Introduction
Let be an n-dimensional simplex in the n-dimensional Euclidean space E n with vertex set
A = {A1 , A2 , . . . , An+1 } and V its volume. The i-th facet fi of is an (n-1)-dimensional
simplex spanned by the vertex set A\{Ai } (i = 1, 2, . . . , n + 1). Let Vi be the content of the
i-th facet fi of , and let Vij denote the volume of the (n-2)-dimensional simplex spanned
by the vertex set A\{Ai , Aj }. Let θi,j denote the internal dihedral angle between fi and
fj (i, j = 1, 2, . . . , n + 1, i = j ).
Let εi (i = 1, 2, . . . , n + 1) be the unit outer normal vector of fi . Denote
Di = (ε1 , . . . , εi−1 , εi+1 , . . . , εn+1 ),
where (ε1 , . . . , εi−1 , εi+1 , . . . , εn+1 ) is the Gram determinant of vectors ε1 , . . . , εi−1 ,
√
εi+1 , . . . , εn+1 , then αi = arcsin Di is called the vertex angle corresponding to the i-th
vertex of (see[1]).
In 1993, Li Hongxiang and Leng Gangson [4] obtained an important inequality as follows.
THEOREM 1. For arbitrary n+1 positive numbers xi (i = 1, 2, . . . , n + 1), we have
n+1 n+1
n
n+1
1
xi sin2 αi ≤
xi
xi−1 .
(1)
n
i=1
i=1
i=1
The equality holds if is a regular simplex and x1 = x2 = · · · = xn+1 .
∗ This work is supported by the Hunan Provincial Science Foundation, P. R. China. The author is a doctorate
candidate of Nanjing University of Science and Technology.
123
124
Triangle inequalities
J. Geom.
This Theorem is also proved in [8].
In this paper, we give a class of triangle inequalities associated with the vertex angles and
dihedral angles of a simplex, which is an improvement of (1). As an application, we improve
an inequality on the volume of the pedal simplex.
THEOREM 2. For the n-dimensional simplex and arbitrary n + 1 nonnegative real
numbers xi (i = 1, 2, . . . , n + 1) in which there are at least n positive numbers, we have


n+1 n
n+1 n−2
n+1
n+1
 
x
)
2(
1
i
i=1
2
2

xj 
xi xj sin θij ≤ n
xi . (2)

 sin αi ≤ (n − 1)nn−1
n
i=1
1≤i<j ≤n+1
j =1
j =i
i=1
In both inequalities, equality holds if is a regular simplex and x1 = x2 = · · · = xn+1 .
For xi > 0 (i = 1, 2, . . . , n + 1), replacing xi by xi−1 in (2), we obtain (1). Therefore the
inequality (2) is an improvement of (1).
2. Some lemmas
To prove the Theorem 2, we need some lemmas.
LEMMA 3.
sin αi =
(nV )n−1
(n − 1)! n+1
j =1 Vj
(i = 1, 2, . . . , n + 1).
(3)
j =i
For the proof of Lemma 3, the reader is referred to [1].
LEMMA 4.
sin θij =
V Vij
n
·
n − 1 Vi Vj
(i, j = 1, 2, . . . , n + 1, i = j ).
(4)
For the proof of Lemma 4, the reader is referred to [3].
LEMMA 5. Let σ = {A1 , A2 , . . . , AN } be a set of N points in E n , N > n and mi
(i = 1, 2, . . . , N ) be positive numbers. Denote by Vi0 i1 ...ik the k-dimensional volume of
the k-dimensional simplex Ai0 Ai1 . . . Aik for every k ∈ {1, 2, . . . , n} and i0 , i1 , . . . , ik ∈
{1, 2, . . . , N}. Put
Mk =
···
i0 <i1 <···<ik
mi0 mi1 . . . mik Vi20 i1 ...ik
M0 = m1 + m2 + · · · + mN ,
(1 ≤ k ≤ n),
Vol. 75, 2002
Sun Mingbao
125
then we have
Mkl
[(n − l)!(l!)3 ]k
(n!M0 )l−k (1 ≤ k < l ≤ n).
[(n − k)!(k!)3 ]l
Mlk
k+1 3 n−k+1
· Mk−1 Mk+1 (1 ≤ k ≤ n).
·
Mk2 ≥
k
n−k
≥
(5)
(6)
Equalities in (5) and (6) hold if and only if the inertial ellipsoid of the points A1 , A2 , . . . , AN
with masses m1 , m2 , . . . , mN is a hypersphere.
For the proof of Lemma 5, the leader is referred to [9], see also [6, XX, 9.17 and 9.18]
LEMMA 6. Let aij = |Ai Aj | (1 ≤ i < j ≤ n + 1) denote the edge-lengths of a simplex
, then
n 1/2
2
2/(n+1)
aij
≥
n!V .
(7)
n+1
1≤i<j ≤n+1
Equality holds if and only if is regular.
For the proof of Lemma 6, the reader is referred to [2].
From Lemma 6, we have
LEMMA 7. Let be a regular simplex with edge-length a, then
V =
1
n!
n+1
2n
1/2
an.
(8)
3. Proof of Theorem 2
For xi > 0(i = 1, 2, . . . , n + 1), from (3) and (4), we get

n+1
i=1




n+1
j =1
 2
xj 
 sin αi =
j =i
n+1
(nV )2(n−1)
n+1
[(n − 1)!]2 (
i=1
Vi )2
i=1

n+1




xj  Vi2 ,

1≤i<j ≤n+1
xi xj sin2 θij =
n2 V 2
n+1
(n − 1)2 ( i=1
Vi )2
1≤i<j ≤n+1
(9)
j =1
j =i

xi xj 

2
n+1
t=1
t=i,j
 2
Vt 
 Vij .
(10)
126
Triangle inequalities
n+1
Taking N = n + 1, mi = (
s=1
s=i
that
n+1
xs )Vi2 (i = 1, 2, . . . , n + 1) in Lemma 5, and observing





xs  =
n+1
i=1
we have


J. Geom.
n+1 n
xi ,
i=1
s=1
s =i

n+1 n n+1 2
2
n+1
 n+1
n+1
2




Mn = 
x
V
V
=
x
Vi V 2
s
i
i


i=1
Mn−1
i=1
s=1
s=i
i=1
(11)
i=1






 2

n+1 n+1
n+1

   n+1



=
xs 
Vi






 i=1
 i=1 j =1 s=1
j =i s=j







2


n+1
n+1
n+1
 n+1
   
x
i



=
xs 
Vi


 n+1 x 



s
s=1
s=1
i=1
j
=1
i=1


s=j
n+1 n−1 n+1 n+1 2
=
xi
xi
Vi .
i=1
i=1

Mn−2 =
1≤i<j ≤n+1

1≤i<j ≤n+1
t=1
s=1
t=i,j

n+1



n+1 n−2
=
xi
i=1
 
2
n+1
  n+1

 n+1
2



xs 
Vt 


 
 Vij

=
(12)
i=1
t=1
n+1



t=1
s=t
s=1
s =t


1≤i<j ≤n+1
2
2
n+1
t=1



 2
Vt 
 Vij
t =i,j

 n+1
2
xi xj 
Vt 

 Vij ,
(13)
t=1
t=i,j

n+1
i=1



xi xj


xs 
 (n+1 x )2 
s=1 s
M0 =
t=i,j

n+1
s=1
s=i
 2
xs 
 Vi .
(14)
Vol. 75, 2002
Sun Mingbao
127
Taking k = n − 1, l = n and k = n − 2, l = n − 1 in inequality (5), respectively, we
obtain
n
Mn−1
Mnn−1
n−1
Mn−2
n−2
Mn−1
≥
n3n−2
M0 ,
[(n − 1)!]2
(15)
≥
[(n − 1)!3 ]n−2 .n!
M0 .
[2(n − 2)!3 ]n−1
(16)
Substituting (11), (12) and (14) into inequality (15), and rearranging the result, we derive


 
n+1 2
n+1
n+1
   2 
n3n−2


(17)
Vi
≥
V 2(n−1) 
xs 


 Vi  .
n+1
[(n − 1)!]2 ( i=1 xi )n
s=1
i=1
i=1
s=i
Substituting (12), (13)and (14) into inequality (16), and rearranging the result, we obtain





1≤i<j ≤n+1
2
n−1

 n+1

2
xi xj 
Vt 

 Vij 
t=1
t =i,j


 
n+1 n−2 n+1 n+1
2(n−2)
   2  n+1
[(n − 1)!3 ]n−2 n! 



≥
xi
xs  Vi 
Vi
.


[2(n − 2)!3 ]n−1
i=1
i=1
(18)
i=1
s=1
s =i
Applying (17) to the right-hand side of (18), and rearranging the result, we get

1≤i<j ≤n+1
≥
2

 n+1
2
xi xj 
Vt 

 Vij
t=1
t=i,j




n+1 n+1

  2
n3(n−2) n!
2(n−2) 


V
xs 


 Vi  .
n−2
x
)
2[(n − 2)!]3 ( n+1
i
i=1
s=1
i=1
(19)
s=i
Dividing (9) by (10), and applying inequality (19), we obtain


n+1 n−2
n+1
n+1
 
2(
i=1 xi )
2

xj 

 sin αi ≤ (n − 1)nn−1
i=1
j =1
j =i
1≤i<j ≤n+1
xi xj sin2 θij .
(20)
128
Triangle inequalities
J. Geom.
Taking N = n + 1, k = n − 1 in inequality (6), we have
2
Mn−1
n
≥2
n−1
3
Mn−2 Mn .
(21)
Substituting (11), (12) and (13) into (21), and rearranging the result, we get
n+1 2 n+1 2
3
n
xi
Vi
≥2
V2
n−1
i=1
i=1

1≤i<j ≤n+1
2

 n+1
2
xi xj 
Vt 

 Vij .
(22)
t=1
t =i,j
Applying (10) and (22), we have
1≤i<j ≤n+1
n+1 2
n−1 xi xj sin θij ≤
xi .
2n
2
(23)
i=1
Applying (20) and (23), we obtain the inequality (2) immediately.
For xi = 0 (i = 1, 2, . . . , n + 1) and xj > 0 (j = 1, 2, . . . , n + 1, j = i), letting xi → 0+
in (2), it turns out that inequality (2) is also true.
If is a regular simplex, applying Lemma 7. to (3) and (4), it is easy to prove that sin αi =
n−n/2 (n + 1)(n−1)/2 (i = 1, 2, . . . , n + 1) and sin θij = n−1 (n2 − 1)1/2 (i, j = 1, 2, . . . ,
n+1, i = j ). Thus equality in (2) holds if is a regular simplex and x1 = x2 = · · · = xn+1 .
4. An application
We keep the notations in Section 1. Let the point P be any internal point of the simplex or one of its facets, and let Hi (i = 1, 2, . . . , n + 1) be the foot of the perpendicular from
P to the (n-1)-dimensional hyperplane spanned by vertex set A\{Ai } of the facet fi of the
simplex . Then the simplex (P ) with {H1 , H2 , . . . , Hn+1 } with vertex set is called the
pedal simplex corresponding to the point P. In particular, if the point P is the incenter I or
circumcenter O of the simplex , it is easy to see that Hi is the tangent point where the
inscribed sphere of touches the facet fi of or the circumcenter Oi of the facet fi ,
then the pedal simplex (P ) is the tangent point simplex (I ) (see[5]) or the circumcenter
simplex (O) of the simplex .
For the pedal simplex (P ), the following conjecture was stated in [7]:
Vol. 75, 2002
Sun Mingbao
129
Let P be any internal point in the simplex , V and Vp denote the volume of the simplex and its pedal simplex (P ), respectively, then
VP ≤
1
V.
nn
(24)
This was first proved in [10], now we give an improvement of inequality (24) by using
Theorem 2 as follows.
THEOREM 8. Let P be any internal point of the simple or one of its facets, and H =
{H1 , H2 , . . . , Hn+1 } be the vertex set of the pedal simplex (P ) with volume VP . Let VHi PH j
denote the volume of the two-dimensional simplex spanned by the points Hi , P , Hj (i, j =
1, 2, . . . , n + 1, i = j ). Then
VP ≤
4
(n − 1)2 nn
Vij VHi P Hj ≤
1≤i<j ≤n+1
1
V.
nn
(25)
In both inequalities, equality holds if is a regular simplex and P is the circumcenter O
of .
REMARK. This theorem seems to be of special interest when P is the incenter or circumcenter of .
Proof of Theorem 8. We consider two cases.
CASE I: the point P is an internal point of the simplex . Denoted by VPi (i = 1, 2, . . . ,
n + 1) the volume of n-dimensional simplex spanned by the points H1 , H2 , . . . , Hi−1 ,
P , Hi+1 . . . , Hn+1 . From the well-known formula for the volume of a simplex, we have
VPi =
−→
−→
−→
−→
−→
−→
1
[(P H 1 , P H 2 , . . . , P H i−1 , P H i+1 , . . . , P H n+1 )]1/2 ,
n!
−→
−→
−→
(26)
−→
where (P H 1 , P H 2 , . . . , P H i−1 , P H i+1 , . . . , P H n+1 ) denotes the Gram determinant
−→
−→
−→
−→
−→
of vectors P H 1 , P H 2 , . . . , P H i−1 , P H i+1 , . . . , P H n+1 .
Applying the definition of the vertex angle αi of simplex in section 1 and (26), we have


VPi =
1 

n! 
n+1
j =1
j =i

|P Hj |
 sinαi .
(27)
130
Triangle inequalities
J. Geom.
Taking xi = |P Hi |Vi (i = 1, 2, . . . , n + 1) in Theorem 2, and using Lemma 3. and (27),
we derive




n+1 n+1
n+1 n+1
  2

 2
n n−1

 sin αi =

x
|P Hj |Vj 
VP .
j



 sin αi = n V
i=1
i=1
j =1
j =i
(28)
j =1
j =i
Applying the geometric theorem for the two-dimensional plane triangle to the two-dimensional
simplex spanned by points Hi , P , Hj , we get
1
|P Hi ||P Hj |sinθij .
(29)
2
Taking xi = |P Hi |Vi (i = 1, 2, . . . , n + 1) in Theorem 2, from Lemma 4. and (29), we
obtain
2n
V
xi xj sin2 θij =
Vij VHi P Hj .
(30)
n−1
VHi PHj =
1≤i<j ≤n+1
1≤i<j ≤n+1
Note the obvious geometric fact
x1 + x2 + · · · + xn = nV .
(31)
Substituting (28), (30) and (31) into inequality (2), we get inequality (25). From the
condition of the equality in inequality (2), we know that the equality in (25) holds if is
regular and the point P is the circumcenter O of .
CASE II: the point P is an internal point of one of the facets fi (i = 1, 2, . . . , n + 1) of
the simplex , taking xi = 0 and xj = |PH j |Vj (j = 1, 2, . . . , n + 1, j = i) in Theorem
2, and repeating the process in the proof of Case I, we prove the assertion. This completes
the proof of Theorem 8.
References
[1] Eriksson, F., The law of sines for tetrahedra and n-simplexes, Geom. Dedicata 7 (1978), 71–80.
[2] Korchmaros, G., Una limitazione per il volume di un simplesso n-dimensional avente spigoli di date
lunghezze, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis Mat. Natar. 56(8) (1974), 876–879.
[3] Leng, G. S. and Tang, L. H., Some inequalities on the inradii of a simplex and its faces, Geom.
Dedicata 61 (1996), 43–49.
[4] Li, H. X. and Leng, G. S., A matrix inequality with weights and its applictions, Linear Algebra Appl. 185
(1993), 273–278.
[5] Mao Qiji and Zou Quanru, A geometric inequality for simplex with tangent points as vertices, Math.
Practice Theory 4 (1987), 71–75 (in Chinese).
[6] Mitrinovic̆, D. S., Pec̆aric̆, J. E. and Volence, V., Recent Advances in Geometric Inequalities, Kluwer Acad.
Publ., Dordrecht, Boston, London, 1989.
[7] Su, H, M., An inequality on simplex, Shuxue Tongbao 5 (1985), 43–46 (in Chinese).
[8] Yang, S. G. and Wang, J., An inequality for n-dimensional sines of vertex angles of a simplex with some
applications, J. Geom. 54 (1995), 198–202.
Vol. 75, 2002
[9]
Sun Mingbao
131
Zhang, J. ZH. and Yang, L., A class of geometric inequalities concerning the mass-points system, J. China
Univ. Sci. Tech. 11(2) (1981), 1–8 (in Chinese).
[10] Zhang, Y., A conjecture on the volume of simplex with feet of perpendicular as vertexes, J. Systems Sci.
Math. Sci. 12 (1992), 371–375 (in Chinese).
Sun Mingbao
Department of Mathematics
Yueyang Normal University
Yueyang 414000, Hunan
P.R. China
and
Department of Applied Mathematics
Nanjing University of Science and Technology
Nanjing 210094, P.R. China
Received 3 July 2000.