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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 131, Number 10, Pages 3039–3042 S 0002-9939(03)06880-1 Article electronically published on May 5, 2003 THE MINIMUM NUMBER OF ACUTE DIHEDRAL ANGLES OF A SIMPLEX GANGSONG LENG (Communicated by Wolfgang Ziller) Abstract. For any n-dimensional simplex Ω ⊂ Rn , we confirm a conjecture of Klamkin and Pook (1988) that there are always at least n acute dihedral angles in Ω. Let Ω be an n-simplex in Rn with vertices A1 , A2 , · · · , An+1 (i.e., Ω = hA1 , A2 , · · · , An+1 i), let Ωi = hA1 , · · · , Ai−1 , Ai+1 , · · · , An+1 i denote its facet which lies in a hyperplane πi , and let ei be the unit outer normal vector of πi (i = 1, 2, · · · , n + 1). Denote by θij the supplement of the angle between ei and ej ; we call θij the dihedral angle between Ωi and Ωj . dihedral angles. In [1], Klamkin Obviously, for any n-simplex, there exist n+1 2 shows that there are always at least three acute dihedral angles in any tetrahedron. Further, he posed the question: can one generalize this result to an n-simplex? Pook [1] conjectures that n is the required minimum number of acute dihedral angles in any n-simplex. In this paper, we confirm this supposition. Theorem 1. There exist at least n acute dihedral angles in any n-simplex, and there exists an n-simplex which has only n acute dihedral angles. It is clear that Theorem 1 can be replaced equivalently by the following statement. Theorem 10 . There exist at most 12 n(n−1) obtuse dihedral angles in any n-simplex, and there exists an n-simplex which has only 12 n(n − 1) obtuse dihedral angles. To prove Theorem 10 , we need the following three lemmas. Lemma 1. Let {x1 , x2 , · · · , xn } be a given linearly independent set of vectors from Rn . Then there exists an n-simplex Ω which takes kxx11 k kxx22 k , · · · , kxxnn k as n unit outer normal vectors of facets. Proof. Since x1 , x2 , · · · , xn are linearly independent, there exist n linearly independent vectors v1 , v2 , · · · , vn such that hxi , vj i = δij kxi k2 (i, j = 1, 2, · · · , n), Received by the editors May 31, 2000 and, in revised form, August 8, 2001. 2000 Mathematics Subject Classification. Primary 52A20. Key words and phrases. Simplex, dihedral angle, dual basis. This work was supported by the National Natural Sciences Foundation of China (10271071). c 2003 American Mathematical Society 3039 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 3040 GANGSONG LENG where δij is the Kronecker delta symbol and h, i denotes the ordinary inner product of Rn . −−−−−→ Without loss of generality, we may assume that vi = An+1 Ai (i = 1, 2, · · · , n). We now consider the n-simplex Ω = hA1 , A2 , · · · , An+1 i. Since xi ⊥ vj (i 6= j), it follows that xi ⊥ πi (i = 1, 2, · · · , n), where πi is the hyperplane spanned by the facet Ωi . This completes the proof of Lemma 1. Lemma 2 (Yang and Zhang [3]). Let Ω and Ω0 be two n-simplices with dihedral 0 (1 ≤ i < j ≤ n + 1), respectively. If angles θij and θij 0 (1 ≤ i < j ≤ n + 1), θij ≤ θij then Ω and Ω0 are similar, namely, 0 (1 ≤ i < j ≤ n + 1). θij = θij Let {u1 , · · · , un } and {s1 , · · · , sn } be two bases of Rn . They are called dual if hui , sj i = δij (1 ≤ i ≤ j ≤ n). It is well known that there exists a unique dual basis for a given basis of Rn . Lemma 3. Let U = {u1 , · · · , un } be a basis of Rn and {s1 , · · · , sn } be the dual basis of U. If hui , uj i < 0 (i 6= j), then hsi , sj i > 0 for all i, j. Proof. We first consider n = 2. Suppose that hu1 , u2 i < 0 and {s1 , s2 } is the dual basis. Then s1 = hs1 , s1 iu1 + hs1 , s2 iu2 . Hence 0 = hs1 , u2 i = hs1 , s1 ihu1 , u2 i + hs1 , s2 i ku2 k2 . Thus hs1 , s2 i > 0. Turning to the case of general n, let H1 = {x ∈ Rn |hu1 , xi = 0}. Then the vectors u2 , · · · , un are on the opposite side of H1 from u1 . For j ≥ 2, let uj = yj + zj , yj ⊥ H1 , zj ∈ H1 ; then yj = hu1 , uj i hu1 , uj i u 1 , zj = u j − u1 . ku1 k2 ku1 k2 If i 6= j and i, j ≥ 2, then hu1 , ui i hu1 , uj i u1 , uj − u1 i ku1 k2 ku1 k2 hu1 , ui ihu1 , uj i hu1 , ui ihu1 , uj i + = hui , uj i − 2 ku1 k2 ku1 k2 hu1 , ui ihu1 , uj i = hui , uj i − < 0. ku1 k2 Note that sk ∈ H1 for k ≥ 2, and hu1 , ui i u1 , sk i = huj , sk i = δjk hzj , sk i = huj − ku1 k2 for k, j ≥ 2. Since {z2 , · · · , zn } and {s2 , · · · , sn } are biorthognal sets of n − 1 vectors in the (n − 1)-dimensional subspace H1 , they must be dual bases in H1 , and hzi , zj i < 0 hzi , zj i = hui − License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ACUTE DIHEDRAL ANGLES OF A SIMPLEX 3041 when i 6= j. If dim H1 > 2, then we can continue projecting to lower-dimensional subspaces until {u1 , · · · , un } is mapped to a basis G = hgn−1 , gn i in a two-dimensional subspace with hgn−1 , gn i < 0 and with {sn−1 , sn } the dual basis of G. But then in dimension 2, hsn−1 , sn i > 0, completing the proof. Proof of Theorem 10 . We will first construct an n-simplex that has exactly 12 n(n−1) obtuse dihedral angles. Denote by Ω an n-simplex all of whose dihedral angles are acute (obviously, such an Ω exists, e.g., regular simplex). Let u1 , u2 , · · · , un be unit outer normal vectors of n facets of Ω. Then hui , uj i < 0, i 6= j, and U = {u1 , · · · , un } is a basis of Rn . Further let {s1 , · · · , sn } be the dual basis of U. By Lemma 1, there exists an n-simplex Ω which takes e1 , · · · , en as unit outer normal vectors of n facets, where ei = kssii k . By Lemma 3, we have hei , ej i = (1.1) 1 hsi , sj i > 0, 1 ≤ i, j ≤ n. ksi k ksj k Let en+1 be the unit outer normal vector of the (n + 1)-th facet of Ω, and fi the (n − 1)-dimensional volume of the facet Ωi of Ω. By Minowski’s realization theorem (see [2, p. 390]), we have n+1 X fi ei = 0. i=1 Therefore, for j ∈ {1, 2, · · · , n}, hen+1 , ej i = h− n X i=1 (1.2) =− n X i=1 ei fi , ej i fn+1 hei , ej i fi fn+1 < 0. By (1.1) and (1.2), we know that Ω has 12 n(n − 1) obtuse dihedral angles and n acute dihedral angles. Assume that there exists an n-simplex whose obtuse dihedral angles are more than 12 n(n − 1). Without loss of generality, we may assume that α1 , · · · , α 12 n(n−1) , α 12 n(n−1)+1 , · · · , α 12 n(n−1)+j are its obtuse dihedral angles, while α 12 n(n−1)+j+1 , · · · , α 12 n(n+1) are acute. Thus, we can construct a rectangular n-simplex with acute dihedral angles α 12 n(n−1)+j+1 , · · · , α 12 n(n+1) . In fact, let {e1 , · · · , en } be a standard base of Rn , and let s en+1 = − 1−λ e1 −· · ·− j s 1−λ ej −cos α 12 n(n−1)+j+1 ej+1 −· · ·−cos α 12 n(n+1) en , j where λ = cos2 α 12 n(n−1)+j+1 + · · · + cos2 α 12 n(n+1) . License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 3042 GANGSONG LENG Then the rectangular simplex that takes e1 , · · · , en , en+1 as n+1 unit normal vectors of facets has n acute dihedral angles s s 1−λ 1−λ , · · · , arccos , α 12 n(n−1)+j+1 , · · · , α 12 n(n+1) . arccos j j By Lemma 2, this is contradictory. This completes the proof. Acknowledgements The author would like to thank the referees for many valuable suggestions and comments. References [1] M. S. Klamkin, L. P. Pook, Acute dihedral angles, Problems 1281, Math. Mag. 61, 5 (1988), 320. [2] R. Schneider, Convex Bodies: the Brunn–Minkowski Theory, Cambridge University Press, Cambridge, 1993. MR 94d:52007 [3] L. Yang and J. Z. Zhang, A necessary and sufficient condition for embedding a simplex with prescribed dihedral angles in E n , Acta Math. Sinica 26 (1983), 250–256. MR 84g:51023 Department of Mathematics, Shanghai University, Shanghai, 200436, People’s Republic of China E-mail address: [email protected] License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use