volume 7, issue 5, article 165, 2006.
Received 18 January, 2005;
accepted 06 June, 2005.
Communicated by:W.S. Cheung
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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
EMail:sxx@ahieedu.net.cn EMail:chenshilong2006@163.com
2000c Victoria University ISSN (electronic): 1443-5756 019-05
Inequalities for Inscribed Simplex and Applications Shiguo Yang and Silong Cheng
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J. Ineq. Pure and Appl. Math. 7(5) Art. 165, 2006
Abstract
In this paper, we study the problem of geometric inequalities for the inscribed simplex of ann-dimensional simplex. An inequality for the inscribed simplex of a simplex is established. Applying it we get a generalization ofn-dimensional Euler inequality and an inequality for the pedal simplex of a simplex.
2000 Mathematics Subject Classification:51K16, 52A40.
Key words: Simplex, Inscribed simplex, Inradius, Circumradius, Inequality.
Contents
1 Main Results . . . 3 2 Some Lemma and Proofs of Theorems. . . 5
References
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1. Main Results
Let σn be an n-dimensional simplex in the n-dimensional Euclidean space En, V denote the volume ofσn, Randrthe circumradius and inradius ofσn, respectively. Let A0, A1, . . . , An be the vertices ofσn, aij = |AiAj|(0 ≤ i <
j ≤n), Fi denote the area of theith facefi =A0· · ·Ai−1Ai+1· · ·An((n−1)- dimensional simplex) ofσn, pointsOandGbe the circumcenter and barycenter of σn, respectively. Fori = 0,1, . . ., n, letA0i be an arbitrary interior point of the ith face fi of σn. The n-dimensional simplexσn0 = A00A01· · ·A0n is called the inscribed simplex of the simplexσn. Leta0ij =|A0iA0j|(0≤i < j ≤n), R0 denote the circumradius ofσ0n, P be an arbitrary interior point ofσn, Pibe the orthogonal projection of the pointP on theith facefiofσn. Then-dimensional simplex σ00n = P0P1· · ·Pn is called the pedal simplex of the pointP with re- spect to the simplex σn [1] – [2], let V00 denote the volume of σn00, R00 andr00 denote the circumradius and inradius ofσn00, respectively. We note that the pedal simplex σ00n is an inscribed simplex of the simplex σn. Our main results are following theorems.
Theorem 1.1. Letσn0 be an inscribed simplex of the simplexσn, then we have (1.1) (R0)2(R2−OG2)n−1 ≥n2(n−1)r2n,
with equality if the simplexσnis regular andσ0nis the tangent point simplex of σn.
Let Ti be the tangent point where the inscribed sphere of the simplex σn touches theith facefiofσn. The simplexσn=T0T1· · ·Tnis called the tangent point simplex of σn [3]. If we takeA0i ≡ Ti (i = 0,1, . . ., n)in Theorem 1.1,
Inequalities for Inscribed Simplex and Applications Shiguo Yang and Silong Cheng
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then σ0n and σn are the same and R0 = r, we get a generalization of the n- dimensional Euler inequality [4] as follows.
Corollary 1.2. For ann-dimensional simplexσn, we have
(1.2) R2 ≥n2r2+OG2,
with equality if the simplexσnis regular.
Inequality (1.2) improves then-dimensional Euler inequality [5] as follows.
(1.3) R≥nr.
Theorem 1.3. Let P be an interior point of the simplex σn, and σn0 the pedal simplex of the pointP with respect toσn, then
(1.4) R00Rn−1 ≥n2n−1(r00)n,
with equality if the simplexσnis regular andσ00nis 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 σ0nbe an inscribed simplex of then-dimensional simplexσn, then we have
(2.1) X
0≤i<j≤n
a0ij2
! n X
i=0
Fi2
!
≥n2(n+ 1)V2,
with equality if the simplexσnis regular andσ0nis the tangent point simplex of σn.
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 simplexσn. Here λi = ViV−1(i = 0,1, . . ., n), Vi is the volume of the simplex σn(i) = BA0· · ·Ai−1Ai+1· · ·AnandPn
i=0λi = 1. LetQbe an arbitrary point inEn, then
−−→ QB =
n
X
i=0
λi−−→
QAi. From this we have
n
X
i=0
λi−−→
BAi =
n
X
i=0
λi−→
QAi−−−→ QB
=−→ 0,
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n
X
i=0
λi−−→
QAi2
=
n
X
i=0
λi−−→
QB+−−→
BAi2
(2.2)
=
n
X
i=0
λi−−→
QB2 + 2−−→ QB·
n
X
i=0
λi−−→
BAi+
n
X
i=0
λi−−→
BAi2
=−−→ QB2+
n
X
i=0
λi−−→
BAi2
.
Forj = 0,1, . . ., n, takingQ≡Aj in (2.2) we get (2.3)
n
X
i=0
λiλj
−−−→ AiAj
2
=λj−−→
BAj2
+λj
n
X
i=0
λi−−→
BAi2
(j = 0,1, . . ., n).
Adding up these equalities in (2.3) and noting thatPn
j=0λj = 1, we get
(2.4) X
0≤i<j≤n
λiλj
−−−→ AiAj
2
=
n
X
i=0
λi
−−→
BAi
2
.
For any real numbersxi > 0 (i = 0,1, . . ., n)and an inscribed simplexσ0n = A00A01· · ·A0nofσn, we take an interior pointB0 ofσn0 such that(λ00, λ01, . . ., λ0n) is the barycentric coordinates of the pointB0with respect to coordinate simplex
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σn0, hereλ0i =xi
.Pn
j=0xj (i= 0,1, . . ., n). Using equality (2.4) we have
X
0≤i<j≤n
λ0iλ0j a0ij2
=
n
X
i=0
λ0i−−→
B0A0i2
,
i.e.
(2.5) X
0≤i<j≤n
xixj a0ij2
=
n
X
i=0
xi
! n X
i=0
xi−−→
B0A0i2! .
SinceB0 is an interior point of σ0nand σ0nis an inscribed simplex ofσn, soB0 is an interior point of σn. Let the pointQi be the orthogonal projection of the pointB0 on theith facefiofσn, then
(2.6)
n
X
i=0
xi−−→
B0A0i2
≥
n
X
i=0
xi−−→
B0Qi2
.
Equality in (2.6) holds if and only ifQi ≡A0i (i= 0,1, . . ., n). In addition, we have
(2.7)
n
X
i=0
−−→B0Qi
Fi =nV.
By the Cauchy’s inequality and (2.7) we have (2.8)
n
X
i=0
xi−−→
B0Qi2
! n X
i=0
x−1i Fi2
!
≥
n
X
i=0
−−→B0Qi ·Fi
!2
= (nV)2.
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Using (2.5), (2.6) and (2.8), we get
(2.9) X
0≤i<j≤n
xixj a0ij2
! n X
i=0
x−1Fi2
!
≥n2
n
X
i=0
xi
! V2.
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 σ0n is the tangent point simplex ofσn.
Lemma 2.2 ([1,6]). For then-dimensional simplexσn, we have
(2.10)
n
X
i=0
Fi2 ≤[nn−4(n!)2(n+ 1)n−2]−1 X
0≤i<j≤n
a2ij
! ,
with equality if the simplexσnis regular.
Lemma 2.3 ([2]). Let P be an interior point of the simplex σ, σ00n the pedal simplex of the pointP with respect toσn, then
(2.11) V ≥nnV00,
with equality if the simplexσnis regular.
Lemma 2.4 ([1]). For then-dimensional simplexσn, we have
(2.12) V ≥ nn/2(n+ 1)(n+1)/2 n! rn,
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Lemma 2.5 ([4]). For then-dimensional simplexσn, we have
(2.13) X
0≤i<j≤n
a2ij = (n+ 1)2
R2−OG2
.
Here O andGare the circumcenter and barycenter of the simplexσn, respec- tively.
Proof of Theorem1.1. Using inequalities (2.1) and (2.10), we get
(2.14) X
0≤i<j≤n
a0ij2
! X
0≤i<j≤n
a2ij
!n−1
≥nn−2(n!)2(n+ 1)n−1V2.
By Lemma2.5we have
(2.15) X
0≤i<j≤n
a0ij2
≤(n+ 1)2(R0)2.
From (2.13), (2.14) and (2.15) we get (2.16) (R0)2
R2−OG2 n−1
≥ nn−1(n!)2 (n+ 1)n+1V2.
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 σ0n is the tangent point simplex ofσn.
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Proof of Theorem1.3. Since the pedal simplexσ00nis an inscribed simplex of the simplexσn, thus inequality (2.16) holds for the pedal simplexσ00n, i.e.
(2.17) (R00)2
R2−OG2n−1
≥ nn−2(n!)2 (n+ 1)n+1V2.
Using inequalities (2.17) and (2.11), we get (2.18) (R00)2R2(n−1) ≥(R00)2
R2−OG2n−1
≥ n3n−2(n!)2 (n+ 1)n+1 (V00)2
By Lemma2.4we have
(2.19) V00 ≥ nn/2(n+ 1)(n+1)/2
n! (r00)n.
From (2.18) and (2.19) we obtain inequality (1.4). It is easy to prove that equal- ity in (1.4) holds if the simplexσnis regular andσn00is the tangent point simplex ofσn.
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References
[1] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´CAND 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 in- equality, J. Geom., 51 (1994), 190–195.
[5] M.S. KLAMKIN, Inequality for a simplex, SIAM. Rev., 27(4) (1985), 576.
[6] L. YANG ANDJ.Zh. ZHANG, A class of geometric inequalities on a finite number of points, Acta Math. Sinica, 23(5) (1980), 740–749.