Journal of Inequalities in Pure and
Applied Mathematics
INEQUALITIES FOR INSCRIBED SIMPLEX AND APPLICATIONS
SHIGUO YANG AND SILONG CHENG
Department of Mathematics
Anhui Institute of Education
Hefei, 230061
P.R. China
volume 7, issue 5, article 165,
2006.
Received 18 January, 2005;
accepted 06 June, 2005.
Communicated by: W.S. Cheung
EMail: sxx@ahieedu.net.cn
EMail: chenshilong2006@163.com
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
019-05
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Abstract
In this paper, we study the problem of geometric inequalities for the inscribed
simplex of an n-dimensional simplex. An inequality for the inscribed simplex of
a simplex is established. Applying it we get a generalization of n-dimensional
Euler inequality and an inequality for the pedal simplex of a simplex.
Inequalities for Inscribed
Simplex and Applications
2000 Mathematics Subject Classification: 51K16, 52A40.
Key words: Simplex, Inscribed simplex, Inradius, Circumradius, Inequality.
Shiguo Yang and Silong Cheng
Contents
1
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Some Lemma and Proofs of Theorems . . . . . . . . . . . . . . . . . . .
References
3
5
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1.
Main Results
Let σn be an n-dimensional simplex in the n-dimensional Euclidean space
E n , V denote the volume of σn , R and r the circumradius and inradius of σn ,
respectively. Let A0 , A1 , . . . , An be the vertices of σn , aij = |Ai Aj | (0 ≤ i <
j ≤ n), Fi denote the area of the ith face fi = A0 · · · Ai−1 Ai+1 · · · An ((n − 1)dimensional simplex) of σn , points O and G be the circumcenter and barycenter
of σn , respectively. For i = 0, 1, . . ., n, let A0i be an arbitrary interior point of
the ith face fi of σn . The n-dimensional simplex σn0 = A00 A01 · · · A0n is called
the inscribed simplex of the simplex σn . Let a0ij = |A0i A0j | (0 ≤ i < j ≤ n), R0
denote the circumradius of σn0 , P be an arbitrary interior point of σn , Pi be the
orthogonal projection of the point P on the ith face fi of σn . The n-dimensional
simplex σn00 = P0 P1 · · · Pn is called the pedal simplex of the point P with respect to the simplex σn [1] – [2], let V 00 denote the volume of σn00 , R00 and r00
denote the circumradius and inradius of σn00 , respectively. We note that the pedal
simplex σn00 is an inscribed simplex of the simplex σn . Our main results are
following theorems.
Theorem 1.1. Let σn0 be an inscribed simplex of the simplex σn , then we have
(1.1)
2
2
(R0 ) (R2 − OG )n−1 ≥ n2(n−1) r2n ,
Inequalities for Inscribed
Simplex and Applications
Shiguo Yang and Silong Cheng
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with equality if the simplex σn is regular and
σn .
σn0
is the tangent point simplex of
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Let Ti be the tangent point where the inscribed sphere of the simplex σn
touches the ith face fi of σn . The simplex σ n = T0 T1 · · · Tn is called the tangent
point simplex of σn [3]. If we take A0i ≡ Ti (i = 0, 1, . . ., n) in Theorem 1.1,
J. Ineq. Pure and Appl. Math. 7(5) Art. 165, 2006
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then σn0 and σ n are the same and R0 = r, we get a generalization of the ndimensional Euler inequality [4] as follows.
Corollary 1.2. For an n-dimensional simplex σn , we have
(1.2)
2
R2 ≥ n2 r2 + OG ,
with equality if the simplex σn is regular.
Inequality (1.2) improves the n-dimensional Euler inequality [5] as follows.
(1.3)
Shiguo Yang and Silong Cheng
R ≥ nr.
Theorem 1.3. Let P be an interior point of the simplex σn , and σn0 the pedal
simplex of the point P with respect to σn , then
(1.4)
Inequalities for Inscribed
Simplex and Applications
n
R00 Rn−1 ≥ n2n−1 (r00 ) ,
with equality if the simplex σn is regular and σn00 is the tangent point simplex of
σn .
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2.
Some Lemma and Proofs of Theorems
To prove the theorems stated above, we need some lemmas as follows.
Lemma 2.1. Let σn0 be an inscribed simplex of the n-dimensional simplex σn ,
then we have
! n
!
X
X
2
(2.1)
a0ij
Fi2 ≥ n2 (n + 1)V 2 ,
i=0
0≤i<j≤n
with equality if the simplex σn is regular and
σn .
σn0
is the tangent point simplex of
Proof. Let B be an interior point of the simplex σn , and (λ0 , λ1 , . . ., λn ) the
barycentric coordinates of the point B with respect to coordinate simplex σn .
Here λi = Vi V −1 (i = 0, 1, .P
. ., n), Vi is the volume of the simplex σn (i) =
BA0 · · · Ai−1 Ai+1 · · · An and ni=0 λi = 1. Let Q be an arbitrary point in E n ,
then
n
−−→ X −−→
QB =
λi QAi .
i=0
From this we have
Inequalities for Inscribed
Simplex and Applications
Shiguo Yang and Silong Cheng
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n
X
i=0
n
−−→ X −→
−−→ −
→
λi BAi =
λi QAi − QB = 0 ,
i=0
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(2.2)
n
X
n
−−→2 X
−−→ −−→2
λi QAi =
λi QB + BAi
i=0
i=0
=
n
X
n
n
i=0
i=0
−−→
−−→ X −−→ X −−→2
λi QB 2 + 2QB ·
λi BAi +
λi BAi
i=0
−−→
= QB 2 +
n
X
−−→2
λi BAi .
i=0
Inequalities for Inscribed
Simplex and Applications
For j = 0, 1, . . ., n, taking Q ≡ Aj in (2.2) we get
(2.3)
n
X
Shiguo Yang and Silong Cheng
−−−→2
λ i λ j A i Aj
i=0
n
−−→2
−−→2
X
= λj BAj + λj
λi BAi
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(j = 0, 1, . . ., n).
i=0
Adding up these equalities in (2.3) and noting that
(2.4)
X
0≤i<j≤n
Pn
j=0 λj = 1, we get
n
−−−→2 X
−−→2
λ i λ j Ai Aj =
λi BAi .
i=0
For any real numbers xi > 0 (i = 0, 1, . . ., n) and an inscribed simplex σn0 =
A00 A01 · · · A0n of σn , we take an interior point B 0 of σn0 such that (λ00 , λ01 , . . ., λ0n )
is the barycentric coordinates of the point B 0 with respect to coordinate simplex
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σn0 , here λ0i = xi
.P
n
j=0
xj (i = 0, 1, . . ., n). Using equality (2.4) we have
X
λ0i λ0j a0ij
2
=
n
X
−−→2
λ0i B 0 A0i ,
i=0
0≤i<j≤n
i.e.
X
(2.5)
xi xj a0ij
2
=
n
X
!
xi
i=0
0≤i<j≤n
n
X
−−→2
xi B 0 A0i
!
.
i=0
Since B 0 is an interior point of σn0 and σn0 is an inscribed simplex of σn , so B 0
is an interior point of σn . Let the point Qi be the orthogonal projection of the
point B 0 on the ith face fi of σn , then
n
X
(2.6)
n
−−→2 X
−−→2
xi B 0 A0i ≥
xi B 0 Qi .
i=0
i=0
A0i
(i = 0, 1, . . ., n). In addition, we
Equality in (2.6) holds if and only if Qi ≡
have
n X
→
−−
(2.7)
B 0 Qi Fi = nV.
i=0
i=0
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By the Cauchy’s inequality and (2.7) we have
! n
!2
!
n
n X
X
X
−−
→2
→
−−
−1 2
0
0
(2.8)
xi B Qi
xi Fi ≥
= (nV )2 .
B Qi · Fi
i=0
Inequalities for Inscribed
Simplex and Applications
i=0
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Using (2.5), (2.6) and (2.8), we get
! n
!
X
X
2
x−1 Fi2 ≥ n2
(2.9)
xi xj a0ij
i=0
0≤i<j≤n
n
X
!
xi V 2 .
i=0
Taking x0 = x1 = · · · = xn = 1 in (2.9), we get inequality (2.1). It is easy
to prove that equality in (2.1) holds if the simplex σn is regular and σn0 is the
tangent point simplex of σn .
Inequalities for Inscribed
Simplex and Applications
Lemma 2.2 ([1, 6]). For the n-dimensional simplex σn , we have
!
n
X
X
2
n−4
2
n−2 −1
2
(2.10)
Fi ≤ [n (n!) (n + 1) ]
aij ,
i=0
Shiguo Yang and Silong Cheng
0≤i<j≤n
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with equality if the simplex σn is regular.
Lemma 2.3 ([2]). Let P be an interior point of the simplex σ,
simplex of the point P with respect to σn , then
V ≥ nn V 00 ,
(2.11)
with equality if the simplex σn is regular.
Lemma 2.4 ([1]). For the n-dimensional simplex σn , we have
(2.12)
V ≥
nn/2 (n + 1)(n+1)/2 n
r ,
n!
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σn00
the pedal
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with equality if the simplex σn is regular.
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Lemma 2.5 ([4]). For the n-dimensional simplex σn , we have
X
2
(2.13)
a2ij = (n + 1)2 R2 − OG .
0≤i<j≤n
Here O and G are the circumcenter and barycenter of the simplex σn , respectively.
Proof of Theorem 1.1. Using inequalities (2.1) and (2.10), we get
(2.14)
X
a0ij
2
X
0≤i<j≤n
Inequalities for Inscribed
Simplex and Applications
!n−1
!
a2ij
≥ nn−2 (n!)2 (n + 1)n−1 V 2 .
By Lemma 2.5 we have
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X
(2.15)
a0ij
2
2
≤ (n + 1)2 (R0 ) .
From (2.13), (2.14) and (2.15) we get
0 2
(R )
2
2 n−1
R − OG
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0≤i<j≤n
(2.16)
Shiguo Yang and Silong Cheng
0≤i<j≤n
nn−1 (n!)2 2
≥
V .
(n + 1)n+1
Using inequalities (2.16) and (2.12), we get inequality (1.1). It is easy to prove
that equality in (1.1) holds if the simplex σn is regular and σn0 is the tangent
point simplex of σn .
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Proof of Theorem 1.3. Since the pedal simplex σn00 is an inscribed simplex of the
simplex σn , thus inequality (2.16) holds for the pedal simplex σn00 , i.e.
2
(R00 )
(2.17)
R2 − OG
2 n−1
≥
nn−2 (n!)2 2
V .
(n + 1)n+1
Using inequalities (2.17) and (2.11), we get
(2.18)
2
2
(R00 ) R2(n−1) ≥ (R00 )
R2 − OG
2 n−1
≥
n3n−2 (n!)2
2
(V 00 )
(n + 1)n+1
By Lemma 2.4 we have
(2.19)
V 00 ≥
Inequalities for Inscribed
Simplex and Applications
Shiguo Yang and Silong Cheng
nn/2 (n + 1)(n+1)/2 00 n
(r ) .
n!
From (2.18) and (2.19) we obtain inequality (1.4). It is easy to prove that equality in (1.4) holds if the simplex σn is regular and σn00 is the tangent point simplex
of σn .
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References
[1] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND V. VOLENEC, Recent Advances
in Geometric Inequalities, Kluwer Acad. Publ., Dordrecht, Boston, London,
1989, 425–552.
[2] Y. ZHANG, A conjecture on the pedal simplex, J. of Sys. Sci. Math. Sci.,
12(4) (1992), 371–375.
[3] G.S. LENG, Some inequalities involving two simplexes, Geom. Dedicata,
66 (1997), 89–98.
[4] Sh.-G. YANG AND J. WANG, Improvements of n-dimensional Euler inequality, J. Geom., 51 (1994), 190–195.
[5] M.S. KLAMKIN, Inequality for a simplex, SIAM. Rev., 27(4) (1985), 576.
[6] L. YANG AND J.Zh. ZHANG, A class of geometric inequalities on a finite
number of points, Acta Math. Sinica, 23(5) (1980), 740–749.
Inequalities for Inscribed
Simplex and Applications
Shiguo Yang and Silong Cheng
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