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Ultrametric Sets in Euclidean Point Spaces The Electronic Journal of Linear Algebra. A publication of the International Linear Algebra Society. Volume 3, pp. 23-30, March 1998. ELA ISSN 1081-3810. ULTRAMETRIC SETS IN EUCLIDEAN POINT SPACES y MIROSLAV FIEDLER Dedicated to Hans Schneider on the occasion of his seventieth birthday. Abstract. Finite sets S of p oints in a Euclidean space the mutual distances of which satisfy the ultrametric inequality A; B maxfA; C ;C; B g for all p oints in S are investigated and geometrically characterized. Inspired by results on ultrametric matrices and previous results on simplices, connections with so-called centered metric trees as well as with strongly isosceles right simplices are found. AMS sub ject classi cations. 51K05, 05C12. Key words. Ultrametric matrices, isosceles, simplex, tree. 1. Intro duction. In the theory of metric spaces cf. [1], a metric is called ultrametric if 1 A; B maxA; C ;C; B holds for all p oints A; B ; C , of the space. Let us observe already at this p oint that if A; B and C are mutually distinct p oints, then they are vertices of an isosceles triangle in which the length of the base do es not exceed the lengths of the legs. This is the reason why ultrametric spaces are also called isosceles spaces. It is well known [5, Theorem 1] that every nite ultrametric space consisting of n + 1 distinct p oints can b e isometrically emb edded into a p oint Euclidean n- dimensional space but not into suchann 1-dimensional space. We reprove this theorem and, in addition, nd a complete geometric characterization of such sets. Supp ose thus that we are given a nite set S of p oints in a p oint Euclidean n- dimensional space. Wesay that this set is ultrametric if the mutual distances satisfy 1 for all triples of p oints in S . In the sequel, we use a result on strictly ultrametric matrices [6], [8]; these are real symmetric and entrywise nonnegative matrices a satisfying ik a mina ;a for all i; j; k ; ik ij jk a > max a for all i: ii ik k 6=i Theorem 1.1. [6], [8]. Every strictly ultrametric matrix A is nonsingular and its inverse B =b is a diagonal ly dominant symmetric M -matrix, i.e. b 0 for ik ik P jb j for al l i. al l i; k , i 6= k , and b > ik ii k 6=i Let us recall now some known facts from the Euclidean distance geometry and qualitative simplex geometry. Fact 1. Let A ;:::;A be points in a Euclidean space E . Then these p oints are 1 m linearly indep endent if and only if T 0 e 2 det 6=0 e M Received by the editors on 14 January 1998. Accepted on 11 March 1998. Handling editor: Daniel Hershkowitz. y Institute of Computer Science Czech Academy of Sciences, Pod vod arenskou vez 2, 182 07 Praha 8, The Czech Republic [email protected]. This researchwas supp orted by grantGACR 201/95/1484. 23 ELA 24 Miroslav Fiedler T where e is a column vector of m ones, e its transp ose and M is the m-by-m Menger matrix the i; k -th entry of whichisthesquare of the distance A ;A . i k Let, for n 2, A ;:::;A , b e linearly indep endent p oints in a p oint Euclidean 1 n+1 n-space E ,thus forming the set of vertices of an n-simplex . By Fact 1, the n + 2- n f by-n + 2 matrix M in the left-hand side of 2 is nonsingular. Consider the matrix e Q =q , r;s =0; 1;:::;n + 1 such that rs 1 e f 3 Q = 2M : This matrix has the following geometric meaning [4]. 2 Fact 2.aq =4r , r b eing the radius of the circumscrib ed sphere to ; 00 1 q are the barycentric co ordinates of the circumcenter of ; b the numb ers 0k 2 c the matrix Q =q , i; k =1;:::;n+ 1 is the Gram matrix the matrix of mutual ik inner pro ducts of the n + 1 outward normals to the n 1-dimensional faces of , normalized so that their sum is the zero vector. Thus, Q is p ositive semide nite with T rank n and the vector 1;:::;1 is -up to a nonzero factor- the only nonzero vector x for which Qx =0. A certain converse also holds as b efore, e is the column vector of ones. Fact 3. Toevery n + 1-by-n + 1 real symmetric p ositive semide nite matrix Q with rank n satisfying Qe = 0 there exists in E , up to congruence in E , a unique n n n-simplex for which the construction ab ove yields this matrix Q. The crucial prop erty of the matrix Q is that it determines the dihedral interior angles of the simplex . Indeed, the dihedral interior angle between the n 1- ik dimensional faces opp osite to A and A i 6= k adds with the angle of the outward i k normals to these faces to so that q ik cos = 4 : p p ik q q ii kk For sake of completeness, we recall and provide a complete answer to the question of characterizing all p ossible distributions of acute, right and obtuse angles in an n- simplex [2]. To make the result more transparent, we transfer these distributions to the edges of : we call the edge A A i 6= k positive if the opp osite dihedral angle i k is acute, the edge will b e called negative if the opp osite angle is obtuse, and the ik edge will b e neutral if the opp osite angle is right. We then refer to signs of the edges the neutral edges will have sign zero. Theorem 1.2. [2] In every n-simplex, the positive edges form a connected set on the set of al l vertices. This condition is also sucient in the sense that whenever we assign signs to edges of an n-simplex in such a way that the positive edges form a connected set on al l vertices, then there exists a position of the vertices that realizes these signs. Proof. Denote N = f1;:::;n +1g. Supp ose that the set of p ositive edges do es not connect the set of vertices fA g, i 2 N . This means that there exists a subset M i of N , ;6= M 6= N , such that in the previous notation, none of the angles , i 2 M , ik k 2 N nM , is acute. By 4, q 0 whenever i 2 M , k 2 N nM . Since Qe =0,we ik P P P have q = 0 for all i.Thus q = 0 which can b e written as ik ik k i2M k X X X q + q =0: ik ik i;k 2M i2M k 2N nM This is imp ossible since the second sum is nonnegative whereas the rst is p ositive: P it is the value of the quadratic form q x x for the vector z =z , z = 1 for ik i k i i i;k ELA Ultrametric sets in Euclidean p oint spaces 25 i 2 M , z = 0 otherwise, and z is non-zero and not a multiple of e. k + To prove the converse, denote by E , E resp ectively, the symmetric subsets of + N N corresp onding to p ositive, resp. negative edges. Supp ose that E connects the set of vertices. The quadratic form X 2 x x i k + i;k ;i<k ;i;k 2E is clearly p ositive semide nite and -b ecause of connectedness- equal to zero only if the + vector x isamultiple of e. The matrix Q of this form is thus p ositive semide nite i + with rank n and Q e = 0. Let similarly Q b e de ned for the analogous quadratic form corresp onding to summing for i; k 2 E . By basic prop erties of p ositive semide nite matrices, all principal minors of order at + most n of Q are p ositive. It follows that for some suciently small p ositive " the + same will b e true for the matrix Q = Q + "Q . Since Qe = 0, this matrix Q satis es the assumptions of Fact 3. The corresp onding simplex will have edges of prescrib ed signs which completes the pro of. Remark 1.3. Observe that we assigned to an n-simplex with n +1 vertices a + signed undirected graph G =N; E ;E whose set of vertices is N and sets of s + p ositive, resp. negative edges are E , E , resp ectively. Here, and in the sequel, we + + omit the neutral edges. By Theorem 1.2, the p ositive subgraph G =N; E is s + connected. By elementary graph theory, the numb er of edges in E is at least n and + in the case that it is n, the graph G is a tree. s Corollary 1.4. [3], [4]. Every n-simplex has at least n acute interior dihedral angles. There exist n-simplices which have exactly n interior dihedral angles acute n and al l the remaining i.e., right. 2 Simplices mentioned in the second part of Corollary 1.4 were called right simplices in [3]. The edges opp osite to acute dihedral angles i.e., p ositive edges were called legs.Thus the legs form a tree, geometrically a set of segments not containing any closed p olygon and connecting the set of vertices of the simplex.
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