JJ II

(1)

volume 5, issue 4, article 106, 2004.

Received 31 July, 2003;

accepted 23 April, 2004.

Communicated by:A. Sofo

Abstract Contents

JJ II

J I

Home Page Go Back

Close Quit

Journal of Inequalities in Pure and Applied Mathematics

THE GENERALIZED SINE LAW AND SOME INEQUALITIES FOR SIMPLICES

SHIGUO YANG

Department of Mathematics Anhui Institute of Education Hefei 230061

People’s Republic of China.

EMail:sxx@ahieedu.net.cn

c

2000Victoria University ISSN (electronic): 1443-5756 106-03

(2)

The Generalized Sine Law and Some Inequalities for Simplices

Shiguo Yang

Title Page Contents

JJ II

J I

Go Back Close

Quit Page2of23

J. Ineq. Pure and Appl. Math. 5(4) Art. 106, 2004

http://jipam.vu.edu.au

Abstract

The sines ofk-dimensional vertex angles of ann-simplex is defined and the law of sines fork-dimensional vertex angles of ann-simplex is established.

Using the generalized sine law forn-simplex, we obtain some inequalities for the sines ofk-dimensional vertex angles of ann-simplex. Besides, we obtain inequalities for volumes ofn-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.

2000 Mathematics Subject Classification:52A40.

Key words: Simplex,k-dimensional vertex angle, Volume, Inequality.

1. Introduction

The law of sines for triangles inE²has natural analogues in higher dimensions.

In 1978, F. Eriksson [1] defined the n-dimensional sines of the n-dimensional corners of ann-simplex inn-dimensional Euclidean spaceEⁿand obtained the law of sines for the n-dimensional corners of ann-simplex. In this paper, the sines of k-dimensional vertex angles of an n -simplex will be defined, and the law of sines fork-dimensional vertex angles of ann-simplex will be established.

Using the generalized sine law for simplices and a known inequality in [2], we get some inequalities for the sines ofk-dimensional vertex angles of ann- simplex.

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 ofn-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].

(4)

Shiguo Yang

Title Page Contents

JJ II

J I

Go Back Close

Quit Page4of23

2. The Generalized Sine Law for Simplices

LetA_i (i= 1,2, . . . , n+ 1)be the vertices of ann-dimensional simplexΩ_nin the n-dimensional Euclidean spaceEⁿ, V the volume of the simplex Ωn and F_i(n−1)-dimensional content of Ω_n. F. Eriksson defined the n-dimensional sines of then-dimensional cornersα_i of then-simplexΩ_nand obtained the law of sines forn-simplices as follows [1]

(2.1) F_i

nsinα_i =

(n−1)!

n+1

Q

j=1

Fj

(nV)ⁿ⁻¹ (i= 1,2, . . . , n+ 1).

In this paper, we will define the sines of the k-dimensional vertex angles of ann-dimensional simplex and establish the law of sines for thek-dimensional vertex angles of an n-simplex. LetV_i₁_i₂···i_k be the(k−1)-dimensional content of the(k−1)-dimensional faceA_i₁A_i₂· · ·A_i_k ((k−1)-simplex) of the simplex Ω_n for k ∈ {2,3, . . . , n + 1} and i₁, i₂, . . . , i_k ∈ {1,2, . . . , n+ 1}, O and R denote the circumcenter and circumradius of the simplex Ω_n respectively.

−→OA_i = Ru_i(i = 1,2, . . . , n+ 1), u_i is the unit vector. The sines of the k- dimensional vertex angles of the simplexΩ_nare defined as follows.

Definition 2.1. Letαij denote the angle formed by the vectorsui anduj. The sine of a k-dimensional vertex angle ϕ_i₁_i₂···i_k of the simplexΩ_ncorresponding the(k−1)-dimensional faceA_i₁A_i₂· · ·A_i_k is defined as

(2.2) sinϕ_i₁_i₂···i_k = (−D_i₁_i₂···i_k)¹²,

(5)

Shiguo Yang

Title Page Contents

JJ II

J I

Go Back Close

Quit Page5of23

where

(2.3) D_i₁_i₂···i_k =

0 1 · · · 1

1

... −¹₂ sin² ^α^ilim₂ 1

(l, m= 1,2, . . . , k).

We will prove that

(2.4) 0<(−D_i₁_i₂···i_k)¹² ≤1.

Ifn= 2, the sine of the 2-dimensional vertex angleϕ_ij of the triangleA₁A₂A₃ is the sine of the angle formed by two edgesA_kA_i andA_kA_j.

With the notation introduced above, we establish the law of sines for the k -dimensional vertex angles of ann-simplex as follows.

Theorem 2.1. For ann-dimensional simplexΩ_ninEⁿandk ∈ {2,3, . . . , n+ 1}, we have

(2.5) V_i₁_i₂···i_k

sinϕ_i₁_i₂···i_k

= (2R)^k−1

(k−1)! (1≤i₁ < i₂ <· · ·< i_k≤n+ 1).

Putϕ12···i−1,i+1,...,n+1 =θ_i, V12···i−1,i+1,...,n+1 =F_i (i= 1,2, . . . , n+ 1), by Theorem2.1we obtain the law of sines for the n-dimensional vertex angles of n-simplices as follows.

Corollary 2.2.

(2.6) F1

sinθ₁ = F2

sinθ₂ =· · ·= Fn+1

sinθ_n+1 = (2R)ⁿ⁻¹ (n−1)!.

(6)

Shiguo Yang

Title Page Contents

JJ II

J I

Go Back Close

Quit Page6of23

If we taken= 2in Theorem2.1or Corollary2.2, we obtain the law of sines for a triangleA1A2A3 in the form

(2.7) a1

sinA₁ = a2

sinA₂ = a3

sinA₃ = 2R.

Proof of Theorem2.1. Letaij =|AiAj|(i, j = 1,2, . . . , n+ 1), then a_ij = 2Rsinα_ij

2 , sin²ϕ_i₁_i₂···i_k

(2.8)

=−D_i₁_i₂···i_k =−

0 1 · · · 1

1

... −_8R¹2a²_i_l_i_m 1

= (−1)^k(8R²)^−(k−1)·

0 1 · · · 1 1

... a²_i

lim

1

(l, m= 1,2, . . . , k).

By the formula for the volume of a simplex, we have sin²ϕ_i₁_i₂···i_k =−D_i₁_i₂···i_k

(2.9)

= (−1)^k(8R²)^−(k−1)(−1)^k2^k−1(k−1)!²V_i²₁_i₂_···i

k

= (k−1)!²

(2R)^2(k−1)V_i²₁_i₂_···i

k.

(7)

Shiguo Yang

Title Page Contents

JJ II

J I

Go Back Close

Quit Page7of23

From this equality (2.5) follows.

Now we prove that inequality (2.4) holds. Whenk ≥2, we havek ≤2^k−1. Using the Voljan-Korchmaros inequality [3], we have

(2.10) V_i₁_i₂···i_k ≤ 1 (k−1)!

k 2^k−1

¹₂

Y

1≤l<m≤k

a_i_l_i_m

!_k² .

Equality holds if and only if the simplexA_i₁A_i₂· · ·A_i_k is regular.

Combining inequality (2.10) with equality (2.5), we get

Vi1i2···i_k ≤ (2R)^k−1 (k−1)! ·

k 2^k−1

¹₂

Y

1≤l<m≤k

sinα_i_l_i_m 2

!²_k (2.11)

≤ (2R)^k−1 (k−1)! ·

k 2^k−1

¹₂

≤ (2R)^k−1 (k−1)!.

Using equality (2.5) and inequality (2.11), we get 0<(−D_i₁_i₂···i_k)¹² = sinϕ_i₁_i₂···i_k = (k−1)!

(2R)^k−1V_i₁_i₂···i_k ≤1.

For the sines of thek-dimensional vertex angles of ann-simplex, we obtain an inequality as follows.

(8)

Shiguo Yang

Title Page Contents

JJ II

J I

Go Back Close

Quit Page8of23

Theorem 2.3. Let ϕ_i₁_i₂···i_k (1 ≤ i₁ < i₂ < · · · < i_k ≤ n + 1) denote the k-dimensional vertex angles of an n-simplex Ωn in Eⁿ, and λi > 0 (i = 1,2, . . . , n+ 1)be arbitrary real numbers, then we have

(2.12) X

1≤i₁<i2<···<i_k≤n+1

λ_i₁λ_i₂· · ·λ_i_ksin²ϕ_i₁_i₂···i_k

≤ n!· Pn+1 i=1 λ_i^k

(n−k+ 1)!(k−1)!(4n)^k−1.

Equality holds ifλ₁ =λ₂ =· · ·=λ_n+1and the simplexΩ_nis regular.

By takingλ₁ =λ₂ =· · ·=λ_n+1in the inequality (2.12), we get:

Corollary 2.4.

(2.13) X

1≤i1<i2<···<i_k≤n+1

sin²ϕi1i2···i_k ≤ n!·(n+ 1)^k

(n−k+ 1)!(k−1)!(4n)^k−1.

Equality holds if the simplexΩ_nis regular.

To prove Theorem2.3, we need a lemma as follows.

Lemma 2.5. LetΩ_nbe ann-simplex inEⁿ, x_i >0 (i= 1,2, . . . , n+ 1)be real numbers, V_i₁_i₂···i_s+1 be the s-dimensional volume of the s-dimensional simplex A_i₁A_i₂· · ·A_i_s+1 fori₁, i₂, . . . , i_s+1 ∈ {1,2, . . . , n+ 1}. Put

M_s = X

1≤i1<i2<···<ik≤n+1

x_i₁x_i₂· · ·x_i_s+1V_i²

1i2···i_s+1, M₀ =

n+1

X

i=1

x_i,

(9)

Shiguo Yang

Title Page Contents

JJ II

J I

Go Back Close

Quit Page9of23

then we have

(2.14) M_s^l≥ [(n−l)!(l!)³]^s

[(n−s)!(s!)³]^l(n!·M₀)^l−sM_l^s(1≤s < l≤n).

Equality holds if and only if the intertial ellipsoid of the pointsA₁, A₂, . . . , A_n+1 with massesx₁, x₂, . . . , x_n+1 is a sphere.

For the proof of Lemma2.5. the reader is referred to [2] or [9].

Proof of Theorem2.3. By puttings= 1, l=k−1andx_i =λ_i(i= 1,2, . . . , n+

1)in the inequality (2.14), we have

(2.15) X

1≤i<j≤n+1

λ_iλ_ja²_ij

!k−1

≥ (n−k+ 1)!·(k−1)!³ [(n−1)!]^k−1 n!·

n+1

X

i=1

λ_i

!^k−2

× X

1≤i1<i2<···<i_k≤n+1

λ_i₁λ_i₂· · ·λ_i_kV_i²

1i2···i_k.

By Theorem2.1, we have

(2.16) V_i₁_i₂···i_k = (2R)^k−1

(k−1)! sinϕ_i₁_i₂···i_k. Using the known inequality [3]

(2.17) X

1≤i<j≤n+1

λ_iλ_ja²_ij ≤

n+1

X

i=1

λ_i

!2

R²,

(10)

Shiguo Yang

Title Page Contents

JJ II

J I

Go Back Close

Quit Page10of23

with equality if and only if the point P = Pn+1

i=1 λ_iA_i is the circumcenter of simplexΩ_n.

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) ifλ₁ = λ₂ = · · · = λ_n+1 and simplex Ωnis regular.

(11)

Shiguo Yang

Title Page Contents

JJ II

J I

Go Back Close

Quit Page11of23

3. Some Inequalities for Volumes of Simplices

Let P be an arbitrary point inside the simplex Ω_n andB_i the orthogonal pro- jection of the point P on the (n −1)-dimensional plane σi containing (n − 1)-simplex f_i = A₁· · ·Ai−1A_i+1· · ·A_n+1. Simplex Ω_n = B₁B₂· · ·B_n+1 is called the pedal simplex of the point P with respect to the simplex Ω_n. Let r_i = |P B_i| (i = 1,2, . . . , n + 1), V be the volume of the pedal simplex Ω_n, V(i)andV(i)denote the volumes of twon-dimensional simplicesΩ_n(i) = A₁· · ·Ai−1P A_i+1· · ·A_n+1 and Ω_n(i) = B₁· · ·Bi−1P B_i+1· · ·B_n+1, respectively. Then we have an inequality for volumes of just defined n-simplices as follows.

Theorem 3.1. LetP be an arbitrary point insiden-dimensional simplexΩ_nand λi(i= 1,2, . . . , n+ 1)positive real numbers, then we have

(3.1)

n+1

X

i=1

λ₁· · ·λ_i−1λ_i+1· · ·λ_n+1V(i)≤ 1 nⁿ

" _n X

i=1

λ_iV(i)

#n

V¹⁻ⁿ,

with equality if the simplex Ω_n is regular, P is the circumcenter of Ω_n and λ₁ =λ₂ =· · ·=λ_n+1.

Now we state some applications of Theorem3.1.

If takingλ1 =λ2 =· · ·=λn+1in inequality (3.1), we have (3.2)

n+1

X

i=1

V(i)≤ 1 nⁿ

"_n+1 X

i=1

V(i)

#n

·V¹⁻ⁿ.

(12)

Shiguo Yang

Title Page Contents

JJ II

J I

Go Back Close

Quit Page12of23

Since the pointP is in the interior of the simplexΩ_n, then (3.3)

n+1

X

i=1

V(i) = V ,

n+1

X

i=1

V(i) =V.

Using (3.2) and (3.3) we obtain an inequality for the volume of the pedal sim- plexΩ_nof the pointP with respect to the simplexΩ_nas follows.

Corollary 3.2. Let P be an arbitrary point inside the n-simplex Ωn, then we have

(3.4) V ≤ 1

nⁿV,

with equality if simplexΩnis regular andP is the circumcenter ofΩn.

Corollary 3.3. Let P be an arbitrary point inside the n-simplex Ω_n, then we have

(3.5)

n+1

X

i=1

V(i)·V(i)≤ 1

(n+ 1)nⁿV²,

with equality if the simplexΩnis regular andP is the circumcenter ofΩn. Proof. Letλ_i = [V(i)]⁻¹(i= 1,2, . . . , n+ 1)in inequality (3.1); we get (3.6)

n+1

X

i=1

V(i)·V(i)≤

n+ 1 n

n

V¹⁻ⁿ

n+1

Y

j=1

V(j).

(13)

Shiguo Yang

Title Page Contents

JJ II

J I

Go Back Close

Quit Page13of23

Using the arithmetic-geometric mean inequality and equality (3.3), we have

n+1

X

i=1

V(i)·V(i)≤

n+ 1 n

n

V¹⁻ⁿ

"

1 n+ 1

n+1

X

j=1

V(j)

#n+1

= 1

(n+ 1)nⁿV². It is easy to see that equality in (3.5) holds if the simplexΩ_n is regular and the pointP is the circumcenter ofΩ_n.

Proof of Theorem3.1. Leth_ibe the altitude of simplexΩ_nfrom vertexA_i, −−→ P B_i

=r_ie_i, wheree_iis the unit outer normal vector of thei-th side facef_i =A₁· · · Ai−1Ai+1· · ·An+1of the simplexΩn, andⁿsinαkbe then-dimensional sine of thek-th cornerα_k of the simplexΩ_n. Wang and Yang [8] proved that

(3.7) ⁿsinα_n= [det(e_i·e_j)_ij6=k]¹² (k= 1,2, . . . , n+ 1).

By the formula for the volume of ann-simplex and (3.7), we have (3.8) V(i) = 1

n![det(r_lr_ke_l·e_k)l,k6=i]¹² = 1 n!







n+1

Y

j=1 j6=i

r_j





·ⁿsinα_i.

Using (3.8), (2.1) andnV =h_iF_i, we get that

n+1

X

i=1

λ₁· · ·λi−1λ_i+1· · ·λ_n+1V(i)

= 1 n!

n+1

X

i=1







n+1

Y

j=1 j6=i

λ_jr_j





·ⁿsinα_i

(14)

Shiguo Yang

Title Page Contents

JJ II

J I

Go Back Close

Quit Page14of23

= 1 n!

n+1

X

i=1













n+1

Y

j=1 j6=i

λ_jr_j





(nV)ⁿ⁻¹





(n−1)!·

n+1

Y

j=1 j6=i

F_j







−1





= [(n!)²·V]⁻¹

n+1

X

i=1







n+1

Y

j=1 j6=i

λ_jr_jh_j





,

i.e.

(3.9) (n!)²V

n+1

X

i=1

λ₁· · ·λi−1λ_i+1· · ·λ_n+1V(i) =

n+1

X

i=1







n+1

Y

j=1 j6=i

λ_jr_jh_j





.

Takings =n−1, l=nin inequality (2.14), we get

(3.10)







n+1

X

i=1







n+1

Y

j=1 j6=i

x_j





F_i²







n

≥ n³ⁿ (n!)²

n+1

X

i=1

x_i

! _n+1 Y

i=1

x_i

!ⁿ⁻¹

V²⁽ⁿ⁻¹⁾.

Letx_i = (λ_ir_ih_i)⁻¹ (i= 1,2, . . . , n+ 1)in inequality (3.10). Then we have

(3.11)

n+1

X

i=1

λ_ir_ih_iF_i²

!n

≥ n³ⁿ (n!)²







n+1

X

i=1







n+1

Y

j=1 j6=i

λ_jr_jh_j





F_i²





·V²⁽ⁿ⁻¹⁾.

(15)

Shiguo Yang

Title Page Contents

JJ II

J I

Go Back Close

Quit Page15of23

Using inequality (3.11) andr_iF_i =nV(i), h_iF_i =nV, we get

(3.12) Vⁿ

"_n+1 X

i=1

λ_iV(i)

#n

≥ nⁿ

(n!)²V²⁽ⁿ⁻¹⁾

n+1

X

i=1







n+1

Y

j=1 j6=i

λ_jr_jh_j





.

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 Ω_n is regular, P is the circumcenter ofΩ_nandλ₁ =λ₂ =· · ·=λ_n+1. Theorem3.1is proved.

Finally, we shall establish some inequalities for volumes of twon-simplices.

As corollaries, the generalizations to several dimensions of the Neuberg-Pedoe inequality and P.Chiakui inequality will be given.

Leta_i (i= 1,2,3)denote the sides of the triangleA₁A₂A₃ with area∆, and a⁰_i(i= 1,2,3)denote the sides of the triangleA⁰₁A⁰₂A⁰₃with area∆⁰, then (3.13)

3

X

i=1

a²_i

3

X

j=1

a⁰_j2

−2 (a⁰_i)²

!

≥16∆∆⁰,

with equality if and only if∆A₁A₂A₃ is similar to∆A⁰₁A⁰₂A⁰₃. Inequality (3.13) is the well-known Neuberg-Pedoe inequality.

In 1984, P. Chiakui [9] proved the following sharpening of the Neuberg-

(16)

Shiguo Yang

Title Page Contents

JJ II

J I

Go Back Close

Quit Page16of23

Pedoe inequality:

(3.14)

3

X

i=1

a²_i

3

X

j=1

a⁰_j2

−2 (a⁰_i)²

!

≥8 (a⁰₁)²+ (a⁰₂)²+ (a⁰₃)²

a²₁ +a²₂ +a²₃ ∆²+ a²₁+a²₂+a²₃

(a⁰₁)²+ (a⁰₂)²+ (a⁰₃)² (∆⁰)²

! ,

with equality if and only if∆A₁A₂A₃ is similar to∆A⁰₁A⁰₂A⁰₃.

Recently, Leng Gangson [5] has extended inequality (3.14) to the edge lengths and volumes of twon-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 Eⁿ, V the volume of the simplex Ω_n and F_i(n−1)- dimensional content of the (n −1)-dimensional face f_i = A₁· · ·Ai−1A_i+1· · ·A_n+1 of Ω_n. For two n-simplicesΩ_nandΩ⁰_nand real numbersα, β ∈(0,1], we put

(3.15) σn(α) =

n+1

X

i=1

F_i^α, σn(β) =

n+1

X

i=1

(F_i⁰)^β, bn= n³ n+ 1

n+ 1 n!²

_n¹ .

We obtain an inequality for volumes of twon-simplices as follows.

Theorem 3.4. For any twon-dimensional simplicesΩ_n andΩ⁰_nand two arbi-

(17)

Shiguo Yang

Title Page Contents

JJ II

J I

Go Back Close

Quit Page17of23

trary real numbersα, β ∈(0,1], we have

(3.16)

n+1

X

i=1

F_i^α

n+1

X

j=1

F_j⁰β

−2 (F_i⁰)^β

!

≥ (n−1)² 2

b^α_nσ_n(β)

σ_n(α)V^2(n−1)α/n+b^β_nσ_n(α)

σ_n(β)(V⁰)^2(n−1)β/n

.

Equality holds if and only if simplicesΩ_nandΩ⁰_nare regular.

Using inequality (3.16) and the arithmetic-geometric mean inequality, we get the following corollary.

Corollary 3.5. For any twon-dimensional simplicesΩ_n andΩ⁰_nand two arbi- trary real numbersα, β ∈(0,1], we have

(3.17)

n+1

X

i=1

F_i^α

n+1

X

j=1

F_j⁰β

−2 (F_i⁰)^β

!

≥b^(α+β)/2_n (n²−1)(V^α(V⁰)^β)^(n−1)/n.

Equality holds if and only if simplicesΩ_nandΩ⁰_nare regular.

If we letα = β in Corollary 3.5, we get Leng Gangson’s inequality [6] as follows. For anyθ ∈(0,1]we have

(3.18)

n+1

X

i=1

F_i^θ

n+1

X

j=1

F_j⁰θ

−2 (F_i⁰)^θ

!

≥b^θ_n(n²−1)(V V⁰)^(n−1)θ/n, with equality if and only if simplicesΩ_nandΩ⁰_nare regular.

To prove Theorem3.4, we need some lemmas as follows.

(18)

Shiguo Yang

Title Page Contents

JJ II

J I

Go Back Close

Quit Page18of23

Lemma 3.6. For ann-simplexΩ_nand arbitrary numberα∈(0,1], we have

(3.19)

Qn+1 i=1 F_i^2α Pn+1

i=1 F_i^2α ≥ 1

(n+ 1)^(n−1)α+1 n³ⁿ

(n!)² α

V^2(n−1)α,

with equality if and only if simplexΩ_nis regular.

Proof. If takingl =n, s=n−1andx_i =F_i²(i= 1,2, . . . , n+1)in inequality (2.14), we get an inequality as follows

(n+ 1)ⁿ(n!)² n³ⁿ

n+1

Y

i=1

F_i² ≥V²⁽ⁿ⁻¹⁾

n+1

X

i=1

F_i²,

or

(3.20) (n+ 1)^nα(n!)^2α n^3nα

n+1

Y

i=1

F_i^2α ≥V^2(n−1)α

n+1

X

i=1

F_i²

!α

.

It is easy to prove that equality in (3.20) holds if and only if simplex Ω_n is regular. From inequality (3.20) we know that inequality (3.19) holds forα= 1.

Forα∈(0,1), using inequality (3.20) and the well-known inequality

(3.21)

n+1

X

i=1

F_i² ≥(n+ 1) 1 n+ 1

n+1

X

i=1

F_i^2α

!¹_α ,

we get inequality (3.19). It is easy to see that equality in (3.19) holds if and only if the simplexΩ_nis regular.

(19)

Shiguo Yang

Title Page Contents

JJ II

J I

Go Back Close

Quit Page19of23

Lemma 3.7. For ann-simplexΩ_n(n≥ 3)and an arbitrary numberα ∈(0,1], we have

(3.22)

n+1

X

i=1

F_i^α

!2

−2

n+1

X

i=1

F_i^2α ≥b^α_n(n²−1)V^2(n−1)α/n,

with equality if and only if the simplexΩ_nis regular.

For the proof of Lemma3.7, the reader is referred to [6].

Lemma 3.8. Let a_i (i = 1,2,3) and ∆ denote the sides and the area of the triangle (A₁A₂A₃), respectively. For arbitrary numberα ∈ (0,1], denote by

∆α the area of the triangle (A1A2A3)α with sides a^α_i (i = 1,2,3), then the following inequality holds

(3.23) ∆²_α ≥ 3

16 16

3 ∆² α

.

Forα6= 1, equality holds if and only ifa₁ =a₂ =a₃. For the proof of Lemma3.8, the reader is referred to [9].

Lemma 3.9. Let numbersx_i >0, y_i >0 (i= 1,2, . . . , n+ 1), σ_n=Pn+1 i=1 x_i, σ_n⁰ =Pn+1

i=1 y_i, then

(3.24) σnσ_n⁰ −2

n+1

X

i=1

xiyi

≥ 1 2

"

σ_n⁰

σ_n σ²_n−2

n+1

X

i=1

x²_i

! +σ_n

σ_n⁰ (σ⁰_n)²−2

n+1

X

i=1

y_i²

!#

,

(20)

Shiguo Yang

Title Page Contents

JJ II

J I

Go Back Close

Quit Page20of23

with equality if and only if y₁ x₁ = y₂

x₂ =· · ·= y_n+1 x_n+1. Proof. Inequality (3.24) is

(3.25) σ_n⁰

σ_n

n+1

X

i=1

x²_i + σ_n σ⁰_n

n+1

X

i=1

y_i² ≥2

n+1

X

i=1

x_iy_i.

Now we prove that inequality (3.25) holds. Using the arithmetic-geometric mean inequality, we have

σ⁰_n

σ_nx²_i +σ_n

σ_n⁰ y_i² ≥2x_iy_i (i= 1,2, . . . , n+ 1).

Adding up thosen+ 1inequalities, we get inequality (3.25). Equality in (3.25) holds if and only if ^σ_σⁿ⁰

nx²_i = ^σ_σⁿ0

ny²_i (i= 1,2, . . . , n+ 1),i.e.

y₁ x₁ = y₂

x₂ =· · ·= y_n+1 x_n+1 = σ⁰_n

σ_n.

Proof of Theorem3.4. Forn = 2, consider two triangles(A₁A₂A₃)_αand(A⁰₁A⁰₂A⁰₃)_β. Using inequality (3.14) and Lemma3.8, we have

(3.26)

3

X

i=1

a^α_i

3

X

j=1

a⁰_jβ

−2 (a⁰_i)^β

!

≥ 1 2

b^α₂σ₂⁰(β)

σ₂(α)∆^α+b^β₂σ₂(α) σ₂⁰(β)(∆⁰)^β

.

(21)

Shiguo Yang

Title Page Contents

JJ II

J I

Go Back Close

Quit Page21of23

Equality in (3.26) holds if and only ifa₁ =a₂ = a₃ anda⁰₁ =a⁰₂ =a⁰₃. Hence, inequality (3.16) holds forn = 2.

Forn ≥ 3, taking x_i =F_i^α, y_i = (F_i⁰)^β (i = 1,2, . . . , n+ 1)in inequality (3.24), we get

n+1

X

i=1

F_i^α

n+1

X

j=1

F_j⁰β

−2 (F_i⁰)^β

! (3.27)

=

n+1

X

i=1

F_i^α

! _n+1 X

i=1

(F_i⁰)^β

!

−2

n+1

X

i=1

F_i^α(F_i⁰)^β

≥ 1 2





 σ_n⁰(β) σn(α)





n+1

X

i=1

F_i^α

!2

−2

n+1

X

i=1

F_i^2α





+σn(α) σ⁰_n(β)





n+1

X

i=1

(F_i⁰)^β

!2

−2

n+1

X

i=1

F_i^2β









 .

Using inequality (3.27) and Lemma3.7, we get

n+1

X

i=1

F_i^α

n+1

X

i=1

F_j⁰β

!

≥ n²−1 2

b^α_nσ_n⁰(β)

σ_n(α)V^2(n−1)α/n+b^β_nσn(α)

σ_n⁰(β)V^2(n−1)β/n

.

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 Ω_n and σ_n⁰ are regular.

Theorem3.4is proved.

(22)

Shiguo Yang

Title Page Contents

JJ II

J I

Go Back Close

Quit Page22of23

References

[1] F. ERIKSSON, The law of sines for tetrahedra andn-simplics, Geometriae Dedicata, 7 (1978), 71–80.

[2] JINGZHONG ZHANG AND LU YANG, A class of geometric inequali- ties 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 di- mensions 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 sim- plices, Geometriae Dedicata, 68 (1997), 43–48.

[6] GANGSON LENG, Some generalizations to several dimensions of the Pe- doe inequality with applications, Acta Mathematica Sinica, 40 (1997), 14–

21.

[7] D. VELJAN, The sine theorem and inequalities for volumes of simplices and determinants, Lin. Alg. Appl., 219 (1995), 79–91.

[8] SHIGUO YANGANDJIA WANG, An inequality forn-dimensional sines of vertex angles of a simplex with some aplications, Journal of Geometry, 54 (1995), 198–202.

(23)

Shiguo Yang

Title Page Contents

JJ II

J I

Go Back Close

Quit Page23of23

[9] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´CANDV. VOLENEC, Recent Geomet- ric Inequalities, Kluwer, Dordrecht, Boston, London, 1989.