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Inequalities for Two Simplices
Journal of Mathematical Analysis and Applications 248, 429᎐437 Ž2000. doi:10.1006rjmaa.2000.6918, available online at http:rrwww.idealibrary.com on
Inequalities for Two Simplices1 Leng Gangsong, Shen Zhu, and Tang Lihua Department of Mathematics, Hunan Normal Uni¨ ersity, Changsha 410081, People’s Republic of China Submitted by B. G. Pachpatte Received December 18, 1998
We establish in this paper an inequality for two simplices, which combine altitudes and edge-lengths of one simplex with distances from an interior point to its facets of the other simplex, and give some applications thereof. 䊚 2000 Academic Press
1. INTRODUCTION The well-known Neuberg᎐Pedoe inequality is the first inequality involving two triangles w14᎐16x. Following Pedoe, a number of inequalities for two triangles have been established w12; 13, XIIx. Yang Lu and Zhang Jingzhong in w18x generalized the Neuberg᎐Pedoe inequality to R n. The research inspired by Yang and Zhang on geometric inequalities for two high-dimensional simplices has been extensive w4, 7᎐10, 17x. The main aim of this paper is to establish a new inequality involving two simplices which combine edge-lengths and altitudes of one simplex with distances from an interior point to its facet of the other simplex. As applications, we obtain some other inequalities for a simplex, and we also sharpen some inequalities of L. Fejes Toth ´ and L. Gerber. We use the following notations throughout this paper. Let ⍀ be an n-dimensional simplex in ⺢ n with vertices A 0 , A1 , . . . , A n Ži.e., ⍀ s ² A 0 , A1 , . . . , A n :. and of volume V, and let I, O, and G be the incenter, circumcenter, and centroid of ⍀, respectively. Let ⍀ i s ² A 0 , . . . , A iy1 , A iq1 , . . . , A n : be its facet which lies in a hyperplane i , Si the facet area of ⍀ i Ži.e., its Ž n y 1.-dimensional volume., d i the distance from an 1
This work is partially supported by the Hunan Provincial Science Foundation. 429 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
430
GANGSONG, ZHU, AND LIHUA
interior point P of ⍀ to i , and h i the altitude of ⍀ from the vertex A i , i.e., the distance from A i to i . Our main results are the following three theorems. THEOREM 1. Let ⍀ s ² A 0 , A1 , . . . , A n : and ⍀X s ² AX0 , AX1 , . . . , AXn : be two n-simplices, h i the altitude of ⍀ from the ¨ ertex A i , a i j s < A i A j < and dXi the distance from an interior point P X of ⍀X to the facet ⍀Xi s ² AX0 , . . . , AXiy1 , AXiq1 , . . . , AXn :. Then dXi dXj
Ý 0Fi-jFn
a2i j
F
1 4
dXi
n
2
žÝ / is1
,
hi
Ž 1.1.
and equality holds if and only if ⍀ is regular and P X is the incenter of ⍀X . Let GX be the centroid of ⍀X and dXi Ž GX . the distance from GX to iX . Taking P X s GX in Theorem 1 and noting the obvious geometric fact that dXi Ž GX . hXi
s
1 nq1
,
we obtain THEOREM 2. Let ⍀ s ² A 0 , A1 , . . . , A n : and ⍀X s ² AX0 , AX1 , . . . , AXn : be two n-simplices with altitudes h 0 , h1 , . . . , h n and hX0 , hX1 , . . . , hXn , respecti¨ ely, a i j s < A i A j < Ž0 F i - j F n.. Then hXi hXj
Ý 0Fi-jFn
a2i j
F
1 4
hXi
n
žÝ / is0
hi
2
,
Ž 1.2.
and equality holds if and only if ⍀ is regular and I X s GX . By Cauchy’s inequality and Ž1.2., we have THEOREM 3. Under the hypotheses in Theorem 2, we ha¨ e n
Ý is0
a2i j hXi hXj
2
G Ž n q 1 . n2
n
hXi
žÝ / is0
hi
y2
,
Ž 1.3.
and equality holds if and only if ⍀ is regular and I X s GX . We will give the proof of Theorem 1 in Section 3, while in Section 2 we will show some applications of the above theorems.
431
INEQUALITIES FOR TWO SIMPLICES
2. SOME INEQUALITIES FOR A SIMPLEX From Theorem 1 we can derive the following two interesting inequalities for a simplex. THEOREM 4. Let d i be the distance from an interior point P of ⍀ s ² A 0 , A1 , . . . , A n : to i Ž i s 0, 1, . . . , n.. Then di d j
Ý 0Fi-jFn
a2i j
Ý
1 4
,
Ž 2.1. 2
di d j
0Fi-jFn
F
a2i j
G n2 Ž n q 1 . ,
Ž 2.2.
and equalities hold if and only if ⍀ is regular and P s I. Proof. Taking ⍀X s ⍀ in Ž1.1. and noting the obvious fact that n
di
Ý is0
s 1,
hi
we obtain the inequality Ž2.1.. By Cauchy’s inequality, we have
ž
a2i j
Ý 0Fi-jFn
di d j
/ž
di d j
Ý
a2i j
0Fi-jFn
G
/ ž
Ž n q 1. n 2
2
/
.
Ž 2.3.
Inequality Ž2.2. follows from Ž2.1. and Ž2.3.. COROLLARY 1. Let d 0 , d1 , . . . , d n be the distances from an interior point P of ⍀ to its Ž n q 1. facets and R the circumradius of ⍀. Then 2r Ž nq1 .
n
ž / Ł di
F
is0
Ý 0Fi-jFn
'd d i
1 n2 j
Ž R 2 y < OG < 2 . ,
F
nq1 2
R,
and equalities hold if and only if ⍀ is regular and P is the center of ⍀.
Ž 2.4. Ž 2.5.
432
GANGSONG, ZHU, AND LIHUA
Proof. According to Ž2.1. and Cauchy’s inequality, we get 1 4
ž
Ý 0Fi-jFn
a2i j G
/ ž
a2i j
Ý 0Fi-jFn
/ž
di d j
Ý
a2i j
0Fi-jFn
2
G
/
ž
Ý 0Fi-jFn
'd d i
j
/
.
Ž 2.6. Noting the well-known formula a2i j s Ž n q 1 .
Ý
2
Ž R 2 y < OG < 2 . ,
Ž 2.7.
0Fi-jFn
from Ž2.6., we derive
Ý
'd d i
0Fi-jFn
j
F
nq1 2
Ž R 2 y < OG < 2 .
1r2
.
Ž 2.8.
Inequalities Ž2.4. and Ž2.5. follow from Ž2.8.. Remark. It is easy to see that Ž2.4. and Ž2.5. are two sharpenings of the Gerber’s inequality w3x, n
Ł di F
is1
R
nq 1
ž /
.
n
Taking P s I in Ž2.4. yields at once that R 2 G Ž nr . q < OG < 2 , 2
which is a sharpening of the famous inequality of L. Fejes Toth ´ w2x: R G nr. This also is an analogy of the interesting inequality of M. S. Klamkin w5x, R 2 G Ž nr . q < OI < 2 . 2
COROLLARY 2. Let h 0 , h1 , . . . , h n be the altitudes of ⍀ s ² A 0 , A1 , . . . , A n :, a i j s < A i A j <. Then hi h j
Ý 0Fi-jFn
a2i j
Ý 0Fi-jFn
F a2i j
hi h j
Ž n q 1. 4 G n2 ,
and equalities hold if and only if ⍀ is regular.
2
,
Ž 2.9. Ž 2.10.
INEQUALITIES FOR TWO SIMPLICES
433
Proof. Inequalities Ž2.9. and Ž2.10. follow by taking ⍀X s ⍀ in Ž1.2. and Ž1.3.. Remark. Taking n s 2 in Ž2.10., we obtain an inequality of Z. Mitrovic, which is regarded as a more elegant inequality for a triangle in w13, Appendix 1, 6.7x. Hence Ž2.10. is a generalization to several dimensions of Mitrovic’s inequality. COROLLARY 3. For an n-dimensional simplex ⍀ s ² A 0 , A1 , . . . , A n :, let a i j s < A i A j <. Then 1
Ý
2 0Fi-jFn a i j
F
1 4r2
,
Ž 2.11.
and equality holds if and only if ⍀ is regular. Proof. Taking for ⍀X in Ž1.2. a regular simplex and noting the known fact that n
Ý is0
1 hi
s
1 r
we obtain Ž2.11.. Remark. The inequality Ž2.11. is a generalization of Walker’s inequality Žsee w13x. to higher-dimensional simplices. For the other proof of Ž2.11. the reader is referred to w7x.
3. THE PROOF OF THEOREM 1 To prove Theorem 1, we need the following two lemmas. LEMMA 1. Let x 0 , x 1 , . . . , x n be Ž n q 1. real constants, and i j Žs ji . denote the internal dihedral angle between facets ⍀ i and ⍀ j of an n-dimensional simplex ⍀. Then n
Ý x i2 G 2 Ý is0
x i x j cos i j ,
Ž 3.1.
0Fi-jFn
and equality holds if and only if Ž x 0 , x 1 , . . . , x n . s Ž S0 , S1 , . . . , Sn . where is any real constant number.
434
GANGSONG, ZHU, AND LIHUA
Proof. Let A s Ž c i j .Ž nq1.=Ž nq1. , where
½
ci j s
if i s j if i / j.
1 ycos i j
Then the matrix A is positive semidefinite w11, 19x. Therefore XAX r G 0
Ž 3.2.
holds for any X s Ž x 0 , x 1 , . . . , x n . g R n. Inequality Ž3.1. follows from Ž3.2.. In the following we show the equality in Ž3.1. holds if and only if X s Ž S0 , S1 , . . . , S n . .
Ž 3.3.
Assume that Ž3.3. holds. Since Žsee w1x. n
Si S j cos i j
Ý Si2 s 2 Ý is0
0Fi-jFn
we have n
XAX r s 2
ž
Si S j cos i j s 0.
Ý Si2 y 2 Ý is0
/
0Fi-jFn
Hence the equality in Ž3.1. holds. Conversely, put n
⌽Ž X . s
Ý x i2 y 2 Ý is0
x i x j cos i j .
0Fi-jFn
Then ⌽ Ž X . G 0 holds for any X g R n. So assume Ž x 0 , x 1 , . . . , x n . g R n such that the equality in Ž3.1. holds, namely ⌽ ŽŽ x 0 , x 1 , . . . , x n .. s 0; then Ž x 0 , x 1 , . . . , x n . is a minimum point of ⌽. Hence we have
⭸⌽ ⭸ xi
s0
Ž i s 0, 1, . . . , n . .
Namely xi y
Ý x j cos i j s 0
Ž i s 0, 1, . . . , n . .
Ž 3.4.
js0 j/i
Equation Ž3.4. can be rewritten in the form ª
AX r s 0.
Ž 3.5.
435
INEQUALITIES FOR TWO SIMPLICES
Since det < A < s 0 and rank A s n Žsee w19x., the system of fundamental solutions of Ž3.5. has only a non-vanishing vector. On the other hand, noting the known fact Žsee w19x. Si s
Ý S j cos i j , js0 j/i
we find that Ž S0 , S1 , . . . , Sn . is a solution vector of Ž3.5.. Hence any solution vector of Ž3.5. satisfies
Ž x 0 , x 1 , . . . , x n . s Ž S0 , S1 , . . . , Sn . , as desired. LEMMA 2 w11x. Let V be the ¨ olume of ⍀ and i j the internal dihedral angle between ⍀ i and ⍀ j , and let Vi j denote the ¨ olume of the Ž n y 2.dimensional simplex ² A 0 , A iy1 , A iq1 , . . . , A jy1 , A jq1 , . . . , A n :. Then Si S j sin i j Vi j
s
nV ny1
.
Proof of Theorem 1. Let Ti j Ž0 F i - j F n. be the bisection of ⍀ at the dihedral angle i j and the Ž n y 1.-dimensional volume < Ti j <, and let Ei j be the intersection of edge A i A j with Ti j . Then ⍀ is divided into two n-simplex ⌺ 1 s ² A 0 , . . . , A iy1 , Ei j , A iq1 , . . . , A n : and ⌺ 2 s ² A 0 , . . . , A iy1 , Ei j , A jq1 , . . . , A n : by Ti j . It follows that V s V Ž ⌺1 . q V Ž ⌺ 2 . ,
Ž 3.6.
where V Ž ⌺1 . and V Ž ⌺ 2 . are the volumes of ⌺ 1 and ⌺ 2 , respectively. By Lemma 2 and Ž3.6., we have
Ž n y 1. nVi j
Si S j sin i j s
Ž n y 1. nVi j
S j < Ti j
i j 2
q
Ž n y 1. nVi j
S i < Ti j
i j 2
.
This clearly implies < Ti j < s
2 Si S j Si q S j
cos
'S S
cos
i j 2
.
Therefore < Ti j < F
i
j
i j 2
.
Ž 3.7.
436
GANGSONG, ZHU, AND LIHUA
From Ž3.7. and Lemma 1, we infer
Ý
dXi dXj < Ti j < 2 F
Ý Ž dXi Si . Ž dXj S j .Ž 1 y cos i j .
1 2
0Fi-jFn
0Fi-jFn 2
n
F
1 4
ž
Ý
dXi Si
is0
/
.
Ž 3.8.
On the other hand, we have a i j < Ti j < s < A i Ei j < < Ti j < q < A j Ei j < < Ti j < G nV Ž ⌺ 1 . q nV Ž ⌺ 2 . s nV.
Ž 3.9.
Combining Ž3.8. and Ž3.9., we obtain
Ž nV .
2
Ý 0Fi-jFn
dXi dXj a2i j
F
1 4
n
2
dXi S i
žÝ /
.
Ž 3.10.
is0
By the formula h i Si s nV, inequality Ž1.1. follows immediately from Ž3.10.. It is clear that equality in Ž1.1. holds if and only if dX0 s dX1 s ⭈⭈⭈ s dXn , S0 s S1 s ⭈⭈⭈ s Sn , and a i j H Ti j . This also is equivalent to saying that P X s GX and ⍀ is regular.
REFERENCES 1. F. Eriksson, The law of sines for tetrahedra and n-simplices, Geom. Dedicata 7 Ž1978., 71᎐80. 2. L. Fejes Toth, ´ Extremum properties of the regular polytopes, Acta Math. Acad. Sci. Hungar. 6 Ž1955., 143᎐146. 3. L. Gerber, The orthocentric simplex as an extreme simplex, Pacific J. Math. 56 Ž1975., 97᎐111. 4. R. J. Gregorac, Then Yang᎐Zhang inequality, J. Geom. 56 Ž1996., 45᎐51. 5. M. S. Klamkin, Problem 85-26, SIAM Re¨ . 27 Ž1985., 576. 6. G. Korchmaros, Una limitazione per il volume di un simplesso n-dimensionale avente spiglo di date lunghezze, Atti Accad. Lincei Rend. 56 Ž1974., 876᎐879. 7. Leng Gangsong, Some inequalities involving two simplices, Geom. Dedicata 66 Ž1997., 89᎐98. 8. Leng Gangsong and Tang Lihua, Some generalizations to several dimensional of the Pedoe inequality with applications, Acta Math. Sinica 40 Ž1997., 14᎐21. 9. Leng Gangsong, Inequalities for edge lengths and volumes of two simplexes, Geom. Dedicata 68 Ž1997., 43᎐48. 10. Leng Gangsong and Qian Xiangzheng, Inequalities for any point and two simplices, Discrete Math. 202 Ž1999., 163᎐172.
INEQUALITIES FOR TWO SIMPLICES
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11. Leng Gangsong and Zhang Yao, The generalized sine theorem and inequalities for simplices, Linear Algebra Appl. 278 Ž1998., 237᎐247. 12. D. S. Mitrinovic and J. E. Pecaric, About the Neuberg᎐Pedoe and the Oppenheim inequalities, J. Math. Anal. Appl. 129 Ž1988., 196᎐210. 13. D. Mitrinovic, J. Pecaric, and V. Volence, ‘‘Recent Advances in Geometric Inequalities,’’ Kluwer Academic, Dordrecht, 1989. 14. D. Pedoe, An inequality for two triangles, Math. Proc. Cambridge Philos. Soc. 38 Ž1942., 397᎐398. 15. D. Pedoe, Thinking geometrically, Amer. Math. Monthly 77 Ž1970., 711᎐721. 16. D. Pedoe, Inside-outside, the Neuberg᎐Pedoe inequality, Uni¨ . Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 544᎐576 Ž1976., 95᎐97. 17. D. Veljan, The sine theorem and inequalities for volumes of simplices and determinants, Linear Algebra Appl. 219 Ž1995., 79᎐91. 18. Yang Lu and Zhang Jingzhong, A generalization to several dimensions of the Neuberg᎐Pedoe inequality with applications, Bull. Austral. Math. Soc. 27 Ž1983., 203᎐214. 19. Yang Lu and Zhang Jingzhong, A sufficient and necessary condition for embedding a simplex with prescribed dihedral angles in E n , Acta Math. Sinica 26 Ž1983., 250᎐256.
Inequalities for Two Simplices1 Leng Gangsong, Shen Zhu, and Tang Lihua Department of Mathematics, Hunan Normal Uni¨ ersity, Changsha 410081, People’s Republic of China Submitted by B. G. Pachpatte Received December 18, 1998
We establish in this paper an inequality for two simplices, which combine altitudes and edge-lengths of one simplex with distances from an interior point to its facets of the other simplex, and give some applications thereof. 䊚 2000 Academic Press
1. INTRODUCTION The well-known Neuberg᎐Pedoe inequality is the first inequality involving two triangles w14᎐16x. Following Pedoe, a number of inequalities for two triangles have been established w12; 13, XIIx. Yang Lu and Zhang Jingzhong in w18x generalized the Neuberg᎐Pedoe inequality to R n. The research inspired by Yang and Zhang on geometric inequalities for two high-dimensional simplices has been extensive w4, 7᎐10, 17x. The main aim of this paper is to establish a new inequality involving two simplices which combine edge-lengths and altitudes of one simplex with distances from an interior point to its facet of the other simplex. As applications, we obtain some other inequalities for a simplex, and we also sharpen some inequalities of L. Fejes Toth ´ and L. Gerber. We use the following notations throughout this paper. Let ⍀ be an n-dimensional simplex in ⺢ n with vertices A 0 , A1 , . . . , A n Ži.e., ⍀ s ² A 0 , A1 , . . . , A n :. and of volume V, and let I, O, and G be the incenter, circumcenter, and centroid of ⍀, respectively. Let ⍀ i s ² A 0 , . . . , A iy1 , A iq1 , . . . , A n : be its facet which lies in a hyperplane i , Si the facet area of ⍀ i Ži.e., its Ž n y 1.-dimensional volume., d i the distance from an 1
This work is partially supported by the Hunan Provincial Science Foundation. 429 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
430
GANGSONG, ZHU, AND LIHUA
interior point P of ⍀ to i , and h i the altitude of ⍀ from the vertex A i , i.e., the distance from A i to i . Our main results are the following three theorems. THEOREM 1. Let ⍀ s ² A 0 , A1 , . . . , A n : and ⍀X s ² AX0 , AX1 , . . . , AXn : be two n-simplices, h i the altitude of ⍀ from the ¨ ertex A i , a i j s < A i A j < and dXi the distance from an interior point P X of ⍀X to the facet ⍀Xi s ² AX0 , . . . , AXiy1 , AXiq1 , . . . , AXn :. Then dXi dXj
Ý 0Fi-jFn
a2i j
F
1 4
dXi
n
2
žÝ / is1
,
hi
Ž 1.1.
and equality holds if and only if ⍀ is regular and P X is the incenter of ⍀X . Let GX be the centroid of ⍀X and dXi Ž GX . the distance from GX to iX . Taking P X s GX in Theorem 1 and noting the obvious geometric fact that dXi Ž GX . hXi
s
1 nq1
,
we obtain THEOREM 2. Let ⍀ s ² A 0 , A1 , . . . , A n : and ⍀X s ² AX0 , AX1 , . . . , AXn : be two n-simplices with altitudes h 0 , h1 , . . . , h n and hX0 , hX1 , . . . , hXn , respecti¨ ely, a i j s < A i A j < Ž0 F i - j F n.. Then hXi hXj
Ý 0Fi-jFn
a2i j
F
1 4
hXi
n
žÝ / is0
hi
2
,
Ž 1.2.
and equality holds if and only if ⍀ is regular and I X s GX . By Cauchy’s inequality and Ž1.2., we have THEOREM 3. Under the hypotheses in Theorem 2, we ha¨ e n
Ý is0
a2i j hXi hXj
2
G Ž n q 1 . n2
n
hXi
žÝ / is0
hi
y2
,
Ž 1.3.
and equality holds if and only if ⍀ is regular and I X s GX . We will give the proof of Theorem 1 in Section 3, while in Section 2 we will show some applications of the above theorems.
431
INEQUALITIES FOR TWO SIMPLICES
2. SOME INEQUALITIES FOR A SIMPLEX From Theorem 1 we can derive the following two interesting inequalities for a simplex. THEOREM 4. Let d i be the distance from an interior point P of ⍀ s ² A 0 , A1 , . . . , A n : to i Ž i s 0, 1, . . . , n.. Then di d j
Ý 0Fi-jFn
a2i j
Ý
1 4
,
Ž 2.1. 2
di d j
0Fi-jFn
F
a2i j
G n2 Ž n q 1 . ,
Ž 2.2.
and equalities hold if and only if ⍀ is regular and P s I. Proof. Taking ⍀X s ⍀ in Ž1.1. and noting the obvious fact that n
di
Ý is0
s 1,
hi
we obtain the inequality Ž2.1.. By Cauchy’s inequality, we have
ž
a2i j
Ý 0Fi-jFn
di d j
/ž
di d j
Ý
a2i j
0Fi-jFn
G
/ ž
Ž n q 1. n 2
2
/
.
Ž 2.3.
Inequality Ž2.2. follows from Ž2.1. and Ž2.3.. COROLLARY 1. Let d 0 , d1 , . . . , d n be the distances from an interior point P of ⍀ to its Ž n q 1. facets and R the circumradius of ⍀. Then 2r Ž nq1 .
n
ž / Ł di
F
is0
Ý 0Fi-jFn
'd d i
1 n2 j
Ž R 2 y < OG < 2 . ,
F
nq1 2
R,
and equalities hold if and only if ⍀ is regular and P is the center of ⍀.
Ž 2.4. Ž 2.5.
432
GANGSONG, ZHU, AND LIHUA
Proof. According to Ž2.1. and Cauchy’s inequality, we get 1 4
ž
Ý 0Fi-jFn
a2i j G
/ ž
a2i j
Ý 0Fi-jFn
/ž
di d j
Ý
a2i j
0Fi-jFn
2
G
/
ž
Ý 0Fi-jFn
'd d i
j
/
.
Ž 2.6. Noting the well-known formula a2i j s Ž n q 1 .
Ý
2
Ž R 2 y < OG < 2 . ,
Ž 2.7.
0Fi-jFn
from Ž2.6., we derive
Ý
'd d i
0Fi-jFn
j
F
nq1 2
Ž R 2 y < OG < 2 .
1r2
.
Ž 2.8.
Inequalities Ž2.4. and Ž2.5. follow from Ž2.8.. Remark. It is easy to see that Ž2.4. and Ž2.5. are two sharpenings of the Gerber’s inequality w3x, n
Ł di F
is1
R
nq 1
ž /
.
n
Taking P s I in Ž2.4. yields at once that R 2 G Ž nr . q < OG < 2 , 2
which is a sharpening of the famous inequality of L. Fejes Toth ´ w2x: R G nr. This also is an analogy of the interesting inequality of M. S. Klamkin w5x, R 2 G Ž nr . q < OI < 2 . 2
COROLLARY 2. Let h 0 , h1 , . . . , h n be the altitudes of ⍀ s ² A 0 , A1 , . . . , A n :, a i j s < A i A j <. Then hi h j
Ý 0Fi-jFn
a2i j
Ý 0Fi-jFn
F a2i j
hi h j
Ž n q 1. 4 G n2 ,
and equalities hold if and only if ⍀ is regular.
2
,
Ž 2.9. Ž 2.10.
INEQUALITIES FOR TWO SIMPLICES
433
Proof. Inequalities Ž2.9. and Ž2.10. follow by taking ⍀X s ⍀ in Ž1.2. and Ž1.3.. Remark. Taking n s 2 in Ž2.10., we obtain an inequality of Z. Mitrovic, which is regarded as a more elegant inequality for a triangle in w13, Appendix 1, 6.7x. Hence Ž2.10. is a generalization to several dimensions of Mitrovic’s inequality. COROLLARY 3. For an n-dimensional simplex ⍀ s ² A 0 , A1 , . . . , A n :, let a i j s < A i A j <. Then 1
Ý
2 0Fi-jFn a i j
F
1 4r2
,
Ž 2.11.
and equality holds if and only if ⍀ is regular. Proof. Taking for ⍀X in Ž1.2. a regular simplex and noting the known fact that n
Ý is0
1 hi
s
1 r
we obtain Ž2.11.. Remark. The inequality Ž2.11. is a generalization of Walker’s inequality Žsee w13x. to higher-dimensional simplices. For the other proof of Ž2.11. the reader is referred to w7x.
3. THE PROOF OF THEOREM 1 To prove Theorem 1, we need the following two lemmas. LEMMA 1. Let x 0 , x 1 , . . . , x n be Ž n q 1. real constants, and i j Žs ji . denote the internal dihedral angle between facets ⍀ i and ⍀ j of an n-dimensional simplex ⍀. Then n
Ý x i2 G 2 Ý is0
x i x j cos i j ,
Ž 3.1.
0Fi-jFn
and equality holds if and only if Ž x 0 , x 1 , . . . , x n . s Ž S0 , S1 , . . . , Sn . where is any real constant number.
434
GANGSONG, ZHU, AND LIHUA
Proof. Let A s Ž c i j .Ž nq1.=Ž nq1. , where
½
ci j s
if i s j if i / j.
1 ycos i j
Then the matrix A is positive semidefinite w11, 19x. Therefore XAX r G 0
Ž 3.2.
holds for any X s Ž x 0 , x 1 , . . . , x n . g R n. Inequality Ž3.1. follows from Ž3.2.. In the following we show the equality in Ž3.1. holds if and only if X s Ž S0 , S1 , . . . , S n . .
Ž 3.3.
Assume that Ž3.3. holds. Since Žsee w1x. n
Si S j cos i j
Ý Si2 s 2 Ý is0
0Fi-jFn
we have n
XAX r s 2
ž
Si S j cos i j s 0.
Ý Si2 y 2 Ý is0
/
0Fi-jFn
Hence the equality in Ž3.1. holds. Conversely, put n
⌽Ž X . s
Ý x i2 y 2 Ý is0
x i x j cos i j .
0Fi-jFn
Then ⌽ Ž X . G 0 holds for any X g R n. So assume Ž x 0 , x 1 , . . . , x n . g R n such that the equality in Ž3.1. holds, namely ⌽ ŽŽ x 0 , x 1 , . . . , x n .. s 0; then Ž x 0 , x 1 , . . . , x n . is a minimum point of ⌽. Hence we have
⭸⌽ ⭸ xi
s0
Ž i s 0, 1, . . . , n . .
Namely xi y
Ý x j cos i j s 0
Ž i s 0, 1, . . . , n . .
Ž 3.4.
js0 j/i
Equation Ž3.4. can be rewritten in the form ª
AX r s 0.
Ž 3.5.
435
INEQUALITIES FOR TWO SIMPLICES
Since det < A < s 0 and rank A s n Žsee w19x., the system of fundamental solutions of Ž3.5. has only a non-vanishing vector. On the other hand, noting the known fact Žsee w19x. Si s
Ý S j cos i j , js0 j/i
we find that Ž S0 , S1 , . . . , Sn . is a solution vector of Ž3.5.. Hence any solution vector of Ž3.5. satisfies
Ž x 0 , x 1 , . . . , x n . s Ž S0 , S1 , . . . , Sn . , as desired. LEMMA 2 w11x. Let V be the ¨ olume of ⍀ and i j the internal dihedral angle between ⍀ i and ⍀ j , and let Vi j denote the ¨ olume of the Ž n y 2.dimensional simplex ² A 0 , A iy1 , A iq1 , . . . , A jy1 , A jq1 , . . . , A n :. Then Si S j sin i j Vi j
s
nV ny1
.
Proof of Theorem 1. Let Ti j Ž0 F i - j F n. be the bisection of ⍀ at the dihedral angle i j and the Ž n y 1.-dimensional volume < Ti j <, and let Ei j be the intersection of edge A i A j with Ti j . Then ⍀ is divided into two n-simplex ⌺ 1 s ² A 0 , . . . , A iy1 , Ei j , A iq1 , . . . , A n : and ⌺ 2 s ² A 0 , . . . , A iy1 , Ei j , A jq1 , . . . , A n : by Ti j . It follows that V s V Ž ⌺1 . q V Ž ⌺ 2 . ,
Ž 3.6.
where V Ž ⌺1 . and V Ž ⌺ 2 . are the volumes of ⌺ 1 and ⌺ 2 , respectively. By Lemma 2 and Ž3.6., we have
Ž n y 1. nVi j
Si S j sin i j s
Ž n y 1. nVi j
S j < Ti j
i j 2
q
Ž n y 1. nVi j
S i < Ti j
i j 2
.
This clearly implies < Ti j < s
2 Si S j Si q S j
cos
'S S
cos
i j 2
.
Therefore < Ti j < F
i
j
i j 2
.
Ž 3.7.
436
GANGSONG, ZHU, AND LIHUA
From Ž3.7. and Lemma 1, we infer
Ý
dXi dXj < Ti j < 2 F
Ý Ž dXi Si . Ž dXj S j .Ž 1 y cos i j .
1 2
0Fi-jFn
0Fi-jFn 2
n
F
1 4
ž
Ý
dXi Si
is0
/
.
Ž 3.8.
On the other hand, we have a i j < Ti j < s < A i Ei j < < Ti j < q < A j Ei j < < Ti j < G nV Ž ⌺ 1 . q nV Ž ⌺ 2 . s nV.
Ž 3.9.
Combining Ž3.8. and Ž3.9., we obtain
Ž nV .
2
Ý 0Fi-jFn
dXi dXj a2i j
F
1 4
n
2
dXi S i
žÝ /
.
Ž 3.10.
is0
By the formula h i Si s nV, inequality Ž1.1. follows immediately from Ž3.10.. It is clear that equality in Ž1.1. holds if and only if dX0 s dX1 s ⭈⭈⭈ s dXn , S0 s S1 s ⭈⭈⭈ s Sn , and a i j H Ti j . This also is equivalent to saying that P X s GX and ⍀ is regular.
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