Triangulations: Applications, Structures, Algorithms

Transcription

1 Triangulations: Applications, Structures, Algorithms Jesús A. De Loera Dept. of Mathematics, University of California Davis, California, USA Jörg Rambau Konrad-Zuse-Zentrum für Informationstechnik Takustr Berlin, Germany rambau@zib.de Francisco Santos Leal Depto. de Matemáticas Estadística y Comp. Universidad de Cantabria Santander, Spain santosf@unican.es December 15, 2004

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3 Contents 1 Motivation: Triangulations in Mathematics Combinatorics and triangulations Optimization and Triangulations Algebra and Triangulations The Rest of this Book Exercises Fundamental Notions How to Construct Triangulations. Regular triangulations Placing, or pushing, triangulation Pulling triangulation Lexicographic triangulations Delaunay triangulations Regular subdivisions Non-generic liftings Lexicographic triangulations, again The set of all subdivisions of a point set Bullet-proof definition of triangulations and subdivisions A more structured approach to subdivisions Subdivisions of point configurations, at last Several equivalent characterizations of triangulations and subdivisions Tools from Convex Geometry Geometric Characterizations Combinatorial Characterizations Hybrid Characterizations Flips and the Graph of Triangulations One step forward. Vector configurations Gale Transforms f -vectors of triangulations A more advanced look at f -vectors of triangulations Notes and References Exercises Life in Two Dimensions Examples of Triangulations Minimum weight triangulations How many Triangulations are there? Upper bound on the number of triangulations All Planar Triangulations are Connected by Flips The Space of Planar Triangulations Optimal Triangulations Notes and References

4 iv Contents Exercises Regular Triangulations and Secondary Polytopes Examples Preliminaries from Polytope Theory The Secondary Polytope and the Secondary Fan The Chamber Fan Flips in the Chamber Fan Notes and References Exercises Non-regular Triangulations The Mother of All Examples, and Some relatives The mother of all examples The double tetrahedron The prism over a simplex The product of two simplices Grid and bipartite graph representations Triangulations of n m A non-regular triangulation of A non-regular triangulation of Cubes and their Slices Cyclic Polytopes Notes and References Exercises A Friendly Space of Triangulations: Cyclic Polytopes Warm-up Example Combinatorial Properties of Cyclic Polytopes Triangulations as Sections of the Canonical Projection Stasheff-Tamari Posets The Structure Theorem for the First Stasheff-Tamari Poset Proof of Claim (i) Notes and References Exercises Unfriendly Spaces of Triangulations Highly Flip-Deficient Triangulations A Construction in Dimension 3 (zig-zag grid) A Construction in Dimension 4 (layers of prisms) Dimension 5: A disconnected graph of triangulations Idea of the construction A locally acyclic orientation in the 24-cell Exponential growth of the number of components Dimension 6: A Triangulation without Flips The Staircase Refinement of a Product The second factor Notes and References

5 Contents v Exercises Enumeration Formulas and Bounds Enumeration of the Regular Component Mathematical Background Algorithms Enumeration of All Triangulations Symmetry-Handling Data Structures and Implementation Details Simplicial Complexes Chirotopes Node in the Flip-Graph Node in the Tree of Partial Triangulations Symmetries Computational Experiments Notes and References Exercises Optimization A Polyhedral Approach: The Universal Polytope Planar Minimum Length Triangulations Other Approaches to the MLT Optimal Size Triangulations of Convex Polytopes The Cube Finding Small Triangulations of 3-polytopes is NP-hard Notes and References Exercises Further Topics Projection-induced subdivisiosn. Fiber polytopes Monotone Paths Zonotopal tilings The general case Mixed subdivisions and he Cayley Trick Mixed subdivisions The Cayley Trick Unimodular Triangulations and Lattice Polytopes Notes and References Exercises

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8 Regular Triangulations 4 and Secondary Polytopes In this chapter we reveal the beautiful structure of the set of regular subdivisions and, in particular, triangulations of a point configuration. We have already seen that a height function w on a point configuration A induces a polyhedral subdivision consisting of a projection of the lower facets of the set A w := { ( a } w(a)) : a A of lifted points. Whenever w is in general position, i.e., no d + 2 lifted points are on a common hyperplane, then we obtain a regular triangulation. For the following, let us denote the polyhedral subdivision induced by a height w by T(A, w). One other structure turns out to be important. Recall from Section?? the notion of refinement: for two polyhedral subdivisions T,T of a point configuration A we say T refines T, in formula T T, if for every cell σ T there is a cell σ T with σ σ. The refinement poset of all polyhedral subdivisions of A, or the Baues-poset of A, denoted by ω(a ), is the set of all polyhedral subdivisions of A, partially ordered by refinement. The subposet of all regular polyhedral subdivisions of A is denoted by ω reg (A ). We will find out in this chapter that ω reg (A ) is remarkably more wellbehaved than ω(a ). Figure 4.1: Example A 1 R 1 Figure 4.2: Alternative heights for T 1 in A 1 Figure 4.3: Alternative heights for T 2 in A Examples Let us get a feeling for regular triangulations by looking at some examples. Figure 4.4: Alternative heights for T 3 in A 1 Example Consider the point configuration A 1 consisting of four points 1,2,4,6 on the real line, labeled 1,2,3,4 from left to right. Here is a list of all triangulations, where the cell {i, j} is denoted by i j for i, j = 1,2,3,4: Figure 4.5: Alternative heights for T 4 in A 1 T 1 := {14}, T 2 := {12,24}, T 3 := {13,34}, T 4 := {12,23,34}. The flip graph of this point configuration is a fourgon, where only T 1 and T 4 and T 2 and T 3 are not connected by a flip. All these triangulations are

9 78 Regular Triangulations and Secondary Polytopes regular. Corresponding heights w Ti are, e.g., given in the following list. Figure 4.6: Heights for T 12 in A 1 w T1 = (1,2,2,1), w T2 = (2,1,2,2), w T3 = (2,2,1,2), w T4 = (2,1,1,2). Figure 4.7: Heights for T 13 in A 1 You will actually prove in Exercise 4.1 that all one-dimensional polyhedral subdivisions are regular. The list of all proper polyhedral subdivisions that are not triangulations reads as follows, where the indices indicate which triangulations refine them: Figure 4.8: Heights for T 24 in A 1 Figure 4.9: Heights for T 34 in A 1 Figure 4.10: Heights for T 1234 in A 1 T 12 := {124}, T 13 := {134}, T 24 := {12,234}, T 34 := {123,34}, T 1234 := {1234}. Again, all of these polyhedral subdivisions are regular. Suitable heights are, e.g., the following (see also Figures 4.6 to 4.10): w T12 := (0,0,1,0), w T13 := (0,1,0,0), w T24 := (1,0,0,0), w T34 := (0,0,0,1), w T1234 := (0,0,0,0). Note that we could have chosen the following alternative heights for the triangulations T 1, T 2, T 3, and T 4 : for each triangulation, pick the coarsest non-trivial polyhedral subdivisions that are strictly refined by it, and sum up their heights w T 1 = w T12 + w T13 = (0,1,1,0), w T 2 = w T12 + w T24 = (1,0,1,0), w T 3 = w T13 + w T34 = (0,1,0,1), w T 4 = w T24 + w T34 = (1,0,0,1). These heights are illustrated in Figures 4.2 to 4.5. That this is a general principle will be the content of Exercise Figure 4.11: Point configurations C 4 Example Consider the pointconfiguration C 4 consisting of the vertices ( 0 0 ), ( 10 ), ( 11 ), and ( 01 ), labeled 1,2,3,4 counter-clockwise (see Figure 4.11).

10 4.1. Examples 79 We find the following polyhedral subdivisions: T 1 := {123,134}, T 2 := {124,234}, T 12 := {1234}. For ε > 0, possible heights are (see Figures 4.12 and 4.13): Figure 4.12: Heights for T 1 in C 4 w 1 := (1 ε,1,1,1), w 2 := (1+ε,1,1,1), w 12 := (1,1,1,1). T 1 and T 2 are connected by a flip. Consider the height homotopy (see Figure 4.14) { A [0,1] R, w : (a,t) (1 t)w 1 (a)+ tw 2 (a) Figure 4.13: Heights for T 2 in C 4 Then, w(a,0) = w 1 (a) and w(a,1) = w 2 (a). Moreover, w(a, 1 2 ) = w 12(a). That means, when we are moving the heights as t goes from 0 to 1, the lower convex hull moves through a singular configuration where the flip region becomes flat. More specifically, in order to obtain T 1 the lifted point 1 must be strictly below the hyperplane spanned by the other three lifted points; in order to obtain T 2 point 1 must be lifted strictly above that hyperplane. In between it will be on that hyperplane. This suggests that flips can somehow be recovered from sliding the height vector across a certain border, at which the height induces a slightly coarser polyhedral subdivision. You have to wait until Section 4.5 for a more specific treatment of this Figure 4.14: A height homotopy simulating a flip in C 4 Example Let us become brave and add another point to the configuration. Let C 5 be the point configuration of the vertices of a convex five-gon. If we lift one of the points very high up, then this will get us a polyhedral subdivision with one triangle and an adjacent four-gon. Refining these polyhedral subdivision will inevitably produce a triangulation. Thus, we have the trivial subdivision plus five subdivisions as described above. Since in each of the non-trivial polyhedral subdivisions the four-gon can be subdivided in two ways, we can count the triangle-triangulation incidences. This yields that every triangle with two edges in the boundary lies in exactly two triangulations, whereas every triangulation contains exactly two triangles with two edge in the boundary. Consequently, since there are five different triangles with two edges in the boundary in C 5, we obtain five triangulations. If we draw the Hasse-diagram of ω(c 5 ) (see Figure 4.15), then we see the Hasse-diagram of another five-gon (see Figure 4.16). Is this a coincidence? Is the reason for this behaviour convexity, dimension two, or the fact that all the polyhedral subdivisions are regular (solve Exercise 4.2 for a Figure 4.15: The Hasse-diagram of ω(c 5 ) Figure 4.16: The face lattice of a five-gon is isomorphic to ω(c 5 )

11 80 Regular Triangulations and Secondary Polytopes Figure 4.17: The mother of all examples A M Figure 4.18: A non-regular triangulation of A M Figure 4.19: Heights for a coarsening of T 1 Figure 4.20: Fixing the height on the middle triangle and on the vertex in the background followed by tweaking the remaining heights (in order to fold along the missing diagonals) leads to a contradiction at the height of the vertex in the background generalization of this fact to arbitrary n-gons)? Stay tuned until Section 4.3, where this question is answered. Besides: did it occur to you that every triangulation of the five-gon contains a core simplex that is contained in no other triangulation? It is the one that has just one edge in the boundary. Once we fix such a triangle, there is a unique triangulation containing it. Exercise 4.10 can be solved by generalizing this idea (did you notice that 5 = 2+3?); solutions that we have in mind will, however, need the methods in Section 4.4. Example (The Mother of All Examples). Consider the point configuration A M R 2 consisting of two nested congruent triangles with parallel edges (see Figure 4.17). In homogeneous coordinates, normalized so that the coordinate sum is four, we can describe A M as follows: A M := { 4 0, 0 4, 0 0, 2 1, 1 2, 1 1 } Let us again label the points in the given order from 1 to 6. The first thing we want to mention is that A M has indeed two non-regular triangulations (T 1 is shown in Figure 4.18): T 1 := {124,245,235,356,346,146,456}, T 2 := {125,145,236,256,134,136,456}. Why are these triangulations non-regular? It is easy to find heights for the polyhedral subdivision T := {1245, 2356, 1346, 456} (see Figure 4.19). In that lifting, where the inner triangle is lifted to a constant height w and the outer triangle is lifted to a constant height w > w, the lower convex hull contains a triangle and three quadrilaterals. That means, the diagonals of the quadrilaterals present in T 1 are not edges of the lower convex hull. Assume now, for the sake of contradiction, that there are heights that induce T 1. We can w.l.o.g. assume that the heights on the inner triangle are again equal to w. Moreover, in order to obtain that triangle as a facet in the lower convex hull, we need strictly larger heights for the outer triangle. They cannot be all equal because then we get the quadrilateral faces as before. In order to get the diagonals as in T 1 into the lower convex hull, we have to lift one higher up than the other. Since the set of diagonals we want to produce is cyclically symmetric, the heights must also satisfy a cyclically symmetric chain of strict inequalities: contradiction (see Figure 4.20). Let us look at the refinement poset of all polyhedral subdivisions of A M. With a little patience, you will find out the following data: there are 18 triangulations and 47 polyhedral subdivisions that are not triangulations (including the trivial one). Moreover, there are 2 non-regular triangulations and 12 non-regular polyhedral subdivisions that are not triangulations. It is actually a good exercise to construct the refinement poset by hand. This example will again teach you that it is significant whether or not an interior point of a cell is considered a member of that cell. For example, the trivial subdivision {123456} has one cell that looks like a triangle but

12 4.2. Preliminaries from Polytope Theory 81 contains as well all the remaining interior points. Therefore, it is not a triangulation. In contrast to this, the polyhedral subdivision {123} contains a cell that looks like the same triangle but does not contain any of the remaining points. Therefore, it is indeed a triangulation. One important fact can be discovered from the refinement poset: the length of maximal chains (number of covering relations in that chain; equal to its number of elements minus one) in that poset is not constant. We find maximal chains of length three and four. In all our previous examples we could assign the polyhedral subdivisions to faces of a polytope such that refinement of subdivisions corresponded to inclusion of faces. But all maximal chains in the face lattice of a polytope have the same length, namely the dimension of the polytope. Therefore, the refinement poset ω(a M ) of A M cannot be isomorphic to a face lattice of any polytope (see Figure 4.21). What is different here? Are the interior points the culprits? Or rather the non-regular polyhedral subdivisions? If we carefully look at the chains of length four, then we notice that all of them contain non-regular polyhedral subdivisions. Therefore, we might still be lucky and find a polytope whose face lattice is isomorphic to the refinement poset ω reg (A M ) of all regular polyhedral subdivisions of A M. So, enough reasons to study this chapter. This, however, will not be possible without some basic knowledge on polytopes and polyhedral cones. 4.2 Preliminaries from Polytope Theory We briefly recall some notions from polytope theory that we need to describe our object of study. See [67] for an extensive treatment. Definition (Positive Hull). The positive hull of a finite set V of vectors in R d is the set cone(v) := { λ v v : λ v 0 v V }. v V The set cone(v) is a convex polyhedral cone. Since we are never dealing with non-convex cones, we will just call cone(v) a polyhedral cone. The dimension of a cone is the dimension of its linear hull. A polyhedral cone is pointed if it does not contain any linear subspace of dimension larger than zero. The lineality space of a polyhedral cone is the largest linear subspace contained in it. A supporting hyperplane of a cone is a hyperplane such that the cone in contained in one of the corresponding closed halfspaces. A face of a cone is the intersection of it with a supporting linear hyperplane. This supporting hyperplane is called a supporting hyperplane of that face.a A proper face of the cone is a face that is neither empty nor the complete cone. The relative interior of a cone is the cone minus the union of its proper faces. We collect some basic facts. Proposition A subset C of R d is a polyhedral cone if and only if it is the intersection of finitely many linear halfspaces, i.e., there exists a finite Figure 4.21: The refinement poset of the mother of all examples A M (rotated by 90 degrees and without most covering-relations; darker points are considered members of the enclosing cell); it contains chains of length three and four

13 82 Regular Triangulations and Secondary Polytopes index set I and φ i (R d ) for i I such that C = { x R d : φ i (x) 0 for all i I }. Proposition Every linear hyperplane { x R d : φ(x) = 0 } = { x R d : φ(x) 0, φ(x) 0 } Figure 4.22: The positive hull of a finite set of vectors Figure 4.23: A polyhedral complex is a polyhedral cone. Proposition The intersection of polyhedral cones is again a polyhedral cone. The following notion of a polyhedral complex is a generalization of a polyhedral subdivision. A polyhedral subdivision of a point configuration can be defined as the set of maximal faces of a polyhedral complex that covers the convex hull and whose vertices are among the points in the point configuration. Definition (Polyhedral Complex). A set P of polyhedrons is a polyhedral complex if (i) P Q P for all P,Q P. (ii) P Q P and P Q Q for all P,Q P. Figure 4.24: Not a polyhedral complex Figure 4.25: A complete pointed fan in dimension two Figure 4.26: Two normally equivalent 5-gons If we use cones for each polyhedron in a polyhedral complex then we obtain a fan. Definition (Fan). A fan in R d is a polyhedral complex consisting of polyhedral cones. A fan is pointed if all of its cones are pointed. It is complete if the union of all its cones covers R d. Definition (Normal Fan). Let P be a polyhedron in R d. Moreover, let F P be a face of P. Then N P (F) := { ψ (R d ) : F ψ = F < P } is a polyhedral cone, the outer normal cone of F in P in R d. For a point p P N P (p) := { ψ (R d ) : ψ, p ψ, p p P } is the outer normal cone of p in P in R d. The corresponding inner normal cones are defined as the negatives of the outer normal cones. The set N P := { N P (F) : F P } = { N P (p) : p P } is the normal fan of P in R d. Two polytopes P and P are normally equivalent if there normal fans are the same: P P : N P = N P.

14 4.3. The Secondary Polytope and the Secondary Fan 83 Proposition Let P be a polytope in R d, not necessarily full-dimensional. Then N P (p) is full-dimensional (i.e., of dimension d), if and only if p is a vertex of P. 4.3 The Secondary Polytope and the Secondary Fan Throughout this section, let A be a full-dimensional point configuration in R d with n points. For any triangulation of A we will construct a vector in R A = R n. The reason why this may be useful will become apparent later. For now, this is just a mysterious definition. Definition (GKZ-Vector). Let T be a triangulation of A. Then φ A (T) := vol(s) e a R A = R n a A σ T :a σ is the Gelfand-Kapranov-Zelevinsky vector of T. Now we have got a finite set of vectors assiciated to our point configuration A. We consider its convex hull. Definition (Secondary Polytope). The set Σ(A ) := conv { φ A (T) : T triangulation of A } R A = R n is the secondary-polytope of A in R n. Why is this polytope interesting? Let us look at our easiest example in order to see some interesting structure. Example (4.1.1 continued). We compute the secondary polytope Σ(A 1 ) of the point configuration A 1 of Example The volumes of all possible simplices can be read off from Figures 4.27 through Figure The GKZ-vectors of the four triangulations are: vol(14) 5 φ A1 (T 1 ) := 0 0 = 0 0 vol(14) 5 vol(12) 1 φ A1 (T 2 ) := vol(12) + vol(24) 0 = 5 0 vol(24) 4 vol(13) 3 φ A1 (T 3 ) := 0 vol(13) + vol(34) = 0 5 vol(34) 2 vol(12) 1 φ A1 (T 4 ) := vol(12) + vol(23) vol(23) + vol(34) = 3 4 vol(34) 2 vol(14) = 5 Figure 4.27: The volumes of simplices in T 1 vol(12) = 1 vol(24) = 4 Figure 4.28: The volumes of simplices in T 2 vol(13) = 3 vol(34) = 2 Figure 4.29: The volumes of simplices in T 3 vol(23) = 2 vol(12) = 1 vol(34) = 2 Figure 4.30: The volumes of simplices in T 4

15 84 Regular Triangulations and Secondary Polytopes Figure 4.31: The refinement poset of A 1 is isomorphic to the face lattice of a four-gon Figure 4.32: The Gauss-Jordan normal form of the non-zero homogeneous coordinates of Σ(A 1 ) in the affine hyperplane with sum of coordinates equal to one Therefore, Σ(A 1 ) is the following convex hull: Σ(A 1 ) = conv { , 5 0, 0 5, 3 } Note that the coordinate sum in every column is ten. This comes as no surprise because the volumes of the simplices in any triangulation must add up to the volume of the point configuration. Every volume of a simplex is counted in the GKZ-vector as many times as this simplex has vertices, and this numbers coincide for all simplices in all triangulations. In the example every volume is counted twice for each triangulation. Since the volume of A 1 is five, the coordinate sum of every GKZ-vector better be equal to ten. In Exercise 4.5, you will find d more affine relations among the coordinates of GKZ-vectors. Can you guess the dimension and the combinatorial structure of Σ(A 1 )? Note that we can arrange all the polyhedral subdivisions in a way that they label the faces of a four-gon and that refinement of polyhedral subdivisions corresponds to face inclusion (see Figure 4.31). May it be that the GKZcoordinates form a four-gon? Let us elaborate on this a little bit. One can calculate by hand or with the help of a computer algebra system that the Gauss-Jordan normal form of the homogeneous coordinates of Σ(A 1 ) is The rank of this matrix is three. Therefore, the affine dimension of Σ(A 1 ) is two. All transformed column vectors lie already in a common hyperplane: they all have coordinate sum equal to one. Moreover, the fourth column is not in the positive hull of the first three since its first coordinate is negative. From this it follows that Σ(A 1 ) is indeed a two-dimensional four-gon embedded in R 4 (see Figure 4.32 for an illustration of the vectors in the Gauss-Jordan normal form, canonically projected into R 3 ). In other words: in this example, the face lattice of Σ(A 1 ) is isomorphic to the refinement poset ω(a 1 ) = ω reg (A 1 ). We will find another description of Σ(A 1 ) in Section 4.4 that makes answering this question without any linear algebra almost trivial. So, stay tuned.

16 4.3. The Secondary Polytope and the Secondary Fan 85 We will now be bold enough to take the outcome of the considerations in Example as evidence for the following yet unproven working hypothesis: the secondary polytope of A has a face lattice isomorphic to the regular refinement poset ω reg (A ) of A. (Remember that we have learned from Example that such a thing cannot be true for ω(a ).) Now, how can we go about actually proving that Σ(A ) always has a structure as in Example 4.3.3? We will see that the face lattice Σ(A ) is related to a natural equivalence structure on height functions: we say that heights w and w on A are equivalent if T(A,w) = T(A,w ). Recall, moreover, that T T if T refines or equals T. This means that every cell of T is contained in some cell of T. Equivalence and refinement will be at the core of our investigations. The following will therefore be our main objects of study for quite some time. Definition (Secondary Cone). For a polyhedral subdivision T of A, let C A (T) := { w R A : T T(A,w) }, C A (T) :={ w R A : T = T(A,w) }. C A (T) is called the secondary cone of T in A, CA (T) is called the relatively open secondary cone of T in A. Although it is not yet justified to call these sets cones, one fact is immediate from the definition: Lemma Let T be a polyhedral subdivision A. Then C A (T) = { C A (T ) : T T }. We show now that the set C A (T) has a very nice description. In the following, we do not aim at the most economic way to develop to the main theorems of this section (Theorems and ). We rather want to collect and prove a companion of facts about the objects of study, together with descriptions of their structures that are as explicit as possible. Let us prepare this by investigating the following question: when is a given polyhedral subdivision T a refinement of the regular subdivision T(A, w)? By definition of refinement, every cell σ T must be contained in a cell σ T(A,w). By definition of T(A,w), this is the case if the lifted cell σ w is contained in a lower facet σ w of the lifted point configuration A w. That means, σ w is coplanar, and all points in A lie weakly above the hyperplane H σ w spanned by σ w. In order to express our geometric considerations in formulas, we need some algebraic expression for the position of points relative to a hyperplane in R d+1. Lying above is the same as lying on the same side as an arbitrary point (for example 0) with infinite height. Now, infinite height seems hard to express in coordinates at first glance. In fact, it is not too hard: the trick is to use homogeneous coordinates of the original points so that all lifted points have (d + 1)nd coordinate equal to one and (d + 2)nd A point x lies on an oriented linear hyperplane in R d spanned by the points x 1,x 2,...,x d if and only if det( ( x ) ( 11, x21 ) (,..., xd1 ) (, x 1) ) = 0. Two points x,y lie on the same side of that hyperplane if and only if signdet( ( ) ( x 11, x21 ) (,..., xd1 ) (, x 1) ) = signdet( ( x ) ( 11, x21 ) (,..., xd1 ) (, y 1) ).

17 86 Regular Triangulations and Secondary Polytopes R ( 00 ) 1 ( 01 ) a ( a1 R d+2 w(a) ) R d+1 ( R d a 1) Figure 4.33: A point with infinite height ( 01 ) like can be equivalently expressed ( 00 ) as in homogeneous coordinates 1 coordinate equal to their heigths, then the point 0 at height + can be easily written as (0,0,1) T (see Figure 4.33). A point x lies on the same side of a linear hyperplane {x : φ(x) = 0 } as a reference point x 0 if and only if φ(x 0 )φ(x) 0. In order to stress the fact that φ(x 0 ) is just used to calibrate the orientation, we will use the equivalent condition sign ( φ(x 0 ) ) φ(x) 0. By standard linear algebra in homogeneous coordinates, φ can be expressed in terms of a determinant function. For every affine basis {s 1,s 2,...,s d+1 } of a cell σ T the resulting constraint on w for σ w being contained in a lower facet of A w reads as follows: [ signdet s 1... s d ] w(s 1 )... w(s d+1 ) 1 det s 1... s d+1 a w(s 1 )... w(s d+1 ) w(a) { = 0 if a σ 0 otherwise The elimination of heights in the sign determinant corresponds to an affine transformation in R d+1 that maps all points ( s ) i w(s i ) in σ w to ( s i Let us call the equality condition the coplanarity condition and the inequality condition the weak folding condition. If we require that the height vector reproduces exactly the polyhedral subdivision we started with then it must satisfy the corresponding strict folding condition, i.e., strict inequality. We claim that the sign of the first determinant does not depend on the height w. Indeed: by adding w(s i ) times the last column to the ith column for i = 1,...,d + 1, we eliminate all the heights to zero without changing anything else. This can be interpreted as follows: if ( a w(a)) is on the same ) side of a hyperplane as ( ) then the oriented (d + 1)-simplex [ ( ) ( ) ( ) s 1 sd+1 a ],...,, w(s 1 ) w(s d+1 ) w(a) has the same orientation as the oriented (d + 1)-simplex [ ( ) ( ) ( ) s 1 sd+1 0 ].,...,, This motivates the following definition. Definition (Lifting Form). Let ρ = {ρ 1,...,ρ d+1 } be an affine basis of A. Then, for an a A the linear form ψ ρ,a (w) := ψ ρ1,...,ρ d+1,a(w) := [ signdet ρ 1... ρ d ] det ρ 1... ρ d+1 a w(ρ 1 )... w(ρ d+1 ) w(a)

18 4.3. The Secondary Polytope and the Secondary Fan 87 is called the lifting form of a w.r.t. ρ. The lifting form ψ ρ,a has the following interpretation: Whenever this linear functional evaluates to zero on some w R A then w lifts the point a onto the hyperplane spanned by the liftings of ρ 1,...,ρ d+1. In this case, ( ρ1 w(ρ 1 ) ),..., ( ρd+1 ) ( w(ρ d+1 ), a w(a)) are coplanar points. Whenever this linear functional evaluates to a positive number then a is lifted above that hyperplane. Thus, we can use the lifting forms to express both the coplanarity and the folding conditions. Before we make this explicit we mention that coplanarity conditions and folding conditions are independent of the choice of the affine basis. (See Figures 4.34 and 4.35 for an illustration.) Lemma Let σ be a full-dimensional subset of A, and let ρ,ρ be affine bases of σ. Then the following hold: (i) Let w R A. Then ψ ρ,a (w) = 0 a σ ψ ρ,a(w) = 0 a σ. (ii) Let w R A so that ψ ρ,a (w) = 0 for all a σ. Then ψ ρ,a (w) = ψ ρ,a(w) a A. Figure 4.34: If a height vector satifies the coplanarity conditions for a cell σ... Figure 4.35:... then the choice of another affine basis of σ produces the same lifiting forms for all points Proof. The assertions follow from the fact that ψ ρ,a (w) = 0 for all a σ means that all a σ lie on the hyperplane spanned by ρ w. It then does not matter which affine basis we choose to describe that hyperplane. Proposition Let w R A. Then the following hold: (i) A cell σ is in T(A,w) if and only if for some, and hence for any affine basis ρ of σ the coplanarity conditions and the strict folding conditions hold, i.e., { = 0 if a σ ψ ρ,a (w) > 0 otherwise (ii) A cell σ is contained in a cell σ in T(A,w) if and only if for some, and hence for any affine basis ρ of σ the coplanarity conditions and the weak folding conditions hold, i.e., { = 0 if a σ ψ ρ,a (w) 0 otherwise (iii) A cell σ is strictly contained in a cell σ in T(A,w) if and only if for some, and hence for any affine basis ρ of σ = 0 if a σ ψ ρ,a (w) = 0 for at least one a A \ σ 0 otherwise

19 88 Regular Triangulations and Secondary Polytopes Proof. The assertions are true by construction of the lifting forms. (They are illustrated in Figures 4.36 to 4.39.) Figure 4.36: The heights on an affine basis of a cell define a hyperplane in R d+1 Figure 4.37: All other members of that cell must be lifted onto that hyperplane Putting together everything yields the following polyhedral description of secondary cones. Corollary Let T be an arbitrary polyhedral subdivision of A. Then (i) The secondary cone of T has the following description: C A (T) = { w R A : ψ ρ,a (w) = 0 ψ ρ,a (w) 0 affine basis ρ σ σ T } a σ a A \ σ = { w R A : ψ ρ,a (w) = 0 a σ ψ ρ,a (w) 0 a A \ σ affine basis for some ρ σ σ T } (ii) The relatively open secondary cone of T has the following description: Figure 4.38: All non-members of that cell must be lifted weakly above that hyperplane; additional points on that hyperplane lead to refinement C A (T) = { w R A : ψ ρ,a (w) = 0 ψ ρ,a (w) > 0 affine basis ρ σ σ T } a σ a A \ σ = { w R A : ψ ρ,a (w) = 0 a σ ψ ρ,a (w) > 0 a A \ σ affine basis for some ρ σ σ T } Proof. The assertions follow from Lemma and Proposition Figure 4.39: If all non-members are lifted strictly above that hyperplane then the heights induce exactly the cell we started with The description above is terribly redundant, i.e., a lot of constraints are implied by sets of other constraints. The description will, however, allow for an easier understanding of faces below. We will discuss a more compact description in Chapter 8. Corollary For all polyhedral subdivisions T of A the secondary cone C A (T) is a polyhedral cone. Next, we compile some basic facts about secondary cones of regular polyhedral subdivisions. In Exercise 4.7 you will seek counterexamples to each assertion when the assumption of regularity is dropped. See Figure 4.40 for a (slightly misleading) illustration.

20 4.3. The Secondary Polytope and the Secondary Fan 89 Proposition Let T,T be regular polyhedral subdivisions of A. Then: (i) C A (T ) is a proper face of C A (T) if and only if T strictly refines T. coarser even coarser (ii) C A (T) is the relative interior of C A (T). heights inducing triangulations (iii) CA (T) is non-empty. (iv) C A (T) is full-dimensional if and only if T is a triangulation. Proof. In order to prove one direction of (i), assume that T and T are regular such that T T. We need to show that is a proper face of C A (T ) = { w R A : ψ ρ,a(w ) = 0 a σ ψ ρ,a(w ) 0 a A \ σ affine basis for some ρ σ σ T } C A (T) = { w R A : ψ ρ,a (w) = 0 a σ ψ ρ,a (w) 0 a A \ σ affine basis for some ρ σ σ T } trivial Figure 4.40: Refinement in a closed secondary cone. In one sense this picture is misleading: the secondary cones in general contain non-trivial lineality spaces. By definition of the secondary cone, C A (T ) C A (T). Thus, C A (T ) = { w C A (T) : ψ ρ,a(w ) = 0 a σ ψ ρ,a(w ) 0 a A \ σ affine basis for some ρ σ σ T } Since T T, we can pick as a basis of σ an affine basis ρ of a cell σ T with σ σ. But then all inequalities of the form ψ ρ,a(w) = ψ ρ,a (w) 0 become redundant because all w C A (T) satisfy them. Therefore, we can define H T := { w R A : ψ ρ,a(w ) = 0 a σ affine basis for some ρ σ σ T }, and C A (T ) = C A (T) H T. In words: C A (T ) just satisfies the additional coplanarity conditions for the cells in T that strictly contain cells of T (see Figure 4.41). This proves that C A (T ) is a face of C A (T).

21 90 Regular Triangulations and Secondary Polytopes additional coplanarity conditions Figure 4.41: The supporting hyperplane for a face enforces the additional coplanarity conditions producing coarser cells In order to show that it is a proper face, we need to show that it is neither empty nor equal to C A (T). Since T is regular, there is a w in R A with T = T(A,w ). This w lies in C A (T ), by definition of the secondary cone. Since T is regular, there is a height vector w in C A (T) with T = T(A,w). Since T T we have in particular T T. Therefore, w is not in C A (T ), by definition of the secondary cone. The reverse direction of the equivalence of Part (i) is obtained by reading the above argument the other way round. From Part (i) this we get immediately (ii), by definition of the relative interior. Assertion (iii) follows from the definition of CA (T). Assertion (iv) follows from the fact that for a regular triangulation there are no coplanarity conditions, i.e., no non-trivial equations, because every cell itself is an affine basis. Therefore, CA (T) is an intersection of finitely many open halfspaces, in particular finitely many open subsets of R A. Therefore, CA (T) is also an open subset of RA. Moreover, it is non-empty since T is regular. Thus, CA (T) is an open, none-empty subset in RA, thus full-dimensional, by elementary metric topology. If, in turn, T is not a triangulation then its secondary cone is a proper face of the secondary cone of a refinement of it, by Part (i). Therefore, its secondary cone cannot be full-dimensional. The following about non-regular polyhedral subdivisions will turn out to be useful as well. Proposition Let T be a non-regular polyhedral subdivision of A. Then: (i) CA (T) is empty. (ii) C A (T) is not full-dimensional. Proof. Assertion (i) is by definition. In order to prove Part (ii) we use Part (i) together with Lemma 4.3.5: C A (T) = { C A (T ) : T T } = { C A (T ) : T regular and T T }. No T in the above union can be a triangulation since a triangulation cannot be strictly refined. Hence, by Proposition (iv), no set in the union is full-dimensional. As a finite union of not full-dimensional cones, C A (T) cannot be full-dimensional. Example (4.1.1 continued). We consider again the point configuration A 1. We show the set of constraints for heights defining the triangulation T 2 = {12,24}. All cells are simplices, so there is no choice for the affine bases. In order to find the simplex 12 in the set of lower facets of a lifting by w 1, w 2, w 3, and w 4, the following conditions must be fulfilled (see Figures 4.42

22 4.3. The Secondary Polytope and the Secondary Fan 91 and 4.43): ψ 1,2,3 (w 1,w 2,w 3,w 4 ) = [ signdet ] det w 1 w 2 1 w 1 w 2 w 3 = ( 2w 1 + 3w 2 w 3 ) > 0, ψ 1,2,4 (w 1,w 2,w 3,w 4 ) = [ signdet ] det w 1 w 2 1 w 1 w 2 w 4 = ( 4w 1 + 5w 2 w 4 ) > 0. The simplex 24 is a lower facet if and only if the following linear constraints are satisfied (see Figures 4.44 and 4.45): ψ 2,4,1 (w 1,w 2,w 3,w 4 ) = [ signdet ] det w 2 w 4 1 w 2 w 4 w 1 = (5w 2 w 4 4w 1 ) > 0, ψ 2,4,3 (w 1,w 2,w 3,w 4 ) = [ signdet ] det w 2 w 4 1 w 2 w 4 w 3 = (2w 2 + 2w 4 4w 3 ) > 0. The complete system of linear equations and equalities specifying the heights that induce T 2 reads as follows (note that two of the above conditions are identical): 2w 1 3w 2 + w 3 > 0 4w 1 5w 2 + w 4 > 0 2w 2 + 4w 3 2w 4 > 0 Since this is the system for a regular triangulation, we have no coplanarity conditions. We see that both the height vectors (2, 1, 2, 2) and (1, 0, 1, 0) in Example indeed fulfill all the constraints. The system of linear equations and equalities specifying the heights that induce subdivisions that are refined by T 2 is the corresponding set of equa- w 3 > 3w 2 2w 1 Figure 4.42: The condition w 3 > 3w 2 2w 1 ensures that the third point is lifted strictly above the line spanned by the lifted first two points; note how the geometric distances among the points in R play an important role as coefficients in the line equations w 4 > 5w 2 4w 1 Figure 4.43: The condition w 4 > 5w 2 4w 1 ensures that the fourth point is lifted strictly above the line spanned by the lifted first two points w 1 > (5w 2 w 4 )/4 Figure 4.44: The condition w 1 > (5w 2 w 4 )/4 ensures that the first point is lifted strictly above the line spanned by the lifted second and fourth point w 3 > (w 4 + w 2 )/2 Figure 4.45: The condition w 3 > (w 2 + w 4 )/2 ensures that the third point is lifted strictly above the line spanned by the lifted second and fourth point

23 92 Regular Triangulations and Secondary Polytopes tions and weak inequalities: 2w 1 3w 2 + w 3 0 4w 1 5w 2 + w 4 0 2w 2 + 4w 3 2w 4 0 common coarsening of four triangulations common coarsening of two triangulations triangulation For example, the height vector (0, 0, 0, 0) fulfills these non-strict inequalities with equality, meaning that T 2 refines the polyhedral subdivision induced by (0,0,0,0), which is the trivial subdivision {1234}. (This is no surprise because every polyhedral subdivision refines the trivial one.) The height vector (1,2,2,1), which induces the triangulation T 1 = {14}, violates all inequalities. Therefore, T 2 does not refine T 1. Mind you again that subdivisions and triangulations are not uniquely determined by the convex hulls of their cells; we need the point sets spanning the cells as well to specify a polyhedral subdivision. triangulation triangulation triangulation common coarsening of all triangulations The following theorem states the conclusion of all our work so far: the collection of secondary cones form a nice subdivision of the space of all possible heights (see Figure 4.46 for a generic sketch). Theorem Let C A be the collection of all secondary cones C A (T) over all regular polyhedral subdivisions T of A. Then C A is a complete polyhedral fan in R d, i.e., (i) For every regular polyhedral subdivision T of A, every face of C A (T) is a member of C A. (ii) C A (T) C A (T ) is a face of both C A (T) and C A (T ). Figure 4.46: The structure of a piece of the secondary fan (intersected with a ball; lineality spaces not drawn): a height in a common face of two or more secondary cones induces a common coarsening of the respective polyhedral subdivisions (iii) The union of all C A (T) over all regular polyhedral subdivisions of A covers R A. Moreover, the face lattice of C A is opposite to the refinement poset of regular polyhedral subdivisions of A. Proof. Assertion (i) follows from Proposition (i). In order to prove (ii), consider the intersection of two cones C A (T) C A (T ). By definition of the secondary cones, this intersection is the set of all heights w R A such that T T(A,w) and T T(A,w). By Proposition (i) again, this means that C A (T) C A (T ) is a face of both C A (T) and C A (T ). Part (iii) follows from the fact that every height vector gives rise to some regular polyhedral subdivision. Proposition (i) finally implies the orderreversing poset isomorphism. Remark The structure of C A in Theorem is related to a certain linear hyperplane arrangement and its cells. The following collection of linear hyperplanes (hyperplanes that contain the origin) forms the linear hyperplane arrangement of liftings of A. (For the experts: in this arrangement, two vectors are in the same cell if they induce the same oriented

24 4.3. The Secondary Polytope and the Secondary Fan 93 matroid on the lifted points.) H ai1,a i2,...,a id+2 := { w R A : ψ ai1,a i2,...,a id+2 (w) = 0, }, H A := { H ai1,a i2,...,a id+2 : a i j A, 1 i j n j = 1,...,d + 2 }. A height w A lies on one of the hyperplanes of H A if it lifts some d + 2 points to an affinely dependent position in R d+1. Such a hyperplane arrangement divides the space R A into cells, and it is known that these cells form a complete polyhedral fan H A in R A (see, e.g., [67]). Unfortunately, cells in H A do not correspond one-to-one to cells in C A : H A is, in general, a refinement of C A. In order to prove Theorem via this correspondence one needs to study the structure of this refinement. A natural question to ask now is the following: is the secondary fan polytopal? That is, does there exist a polytope whose normal fan it is? Here, at last, the secondary polytope from Definition enters the scene. In the remaining part of this section we will show that it provides the answer to the polytopality question. To this end, we now turn our attention to a representation in R d+1 of any triangulation with respect to a given height function w. The key observation is that any map on the vertices of a polyhedral subdivision can be extended affinely on the interior of the higher dimensional cells. In order to see the connection of the secondary fan and the secondary polytope, we need to understand the equivalence and refinement relations on heights from the perspective of liftings. See Figure 4.47 for an illustration. Definition (Lifting Function). Let T be a polyhedral subdivision of A and let w : A R be a height function. Then { R d R g w,t : d+1, a g w,t (a) := w(a), extended affinely on each cell σ T, is called the lifting function of T w.r.t. w. We remark that this is well-defined on intersections of simplices because of the intersection property of a polyhedral subdivision. In order to relate lifting functions to things we already know we list an observation: Lemma The function graph G w,t(a,w) := { ( ) x : x conva } R d+1 g w,t(a,w) (x) Figure 4.47: The graph of the piecewise affine function defined by a triangulation and a height vector. In this case the function is as convex as possible. Thus, the height induces the given triangulation equals the union of lower facets of A w. So, if T = T(A,w) then we just get a formaly different description of the definition of T(A,w). Note, however, that in the definition of g w,t we do

25 94 Regular Triangulations and Secondary Polytopes G w1,t 4 G w1,t 3 G w1,t 2 G w1,t 1 Figure 4.48: The lifting functions corresponding to w 1 = (1,2,2,1): the one correspoinding to T 1 is minimal G w2,t 1 G w2,t 3 G w2,t 4 G α2,t 2 Figure 4.49: The lifting functions corresponding to w 2 = (2,1,2,2): the one correspoinding to T 2 is minimal G α3,t 1 G w3,t 2 G w3,t 4 G w3,t 3 Figure 4.50: The lifting functions corresponding to w 3 = (2,2,1,2): the one correspoinding to T 3 is minimal G w4,t 1 G w4,t 3 G w4,t 2 G w4,t 4 Figure 4.51: The lifting functions corresponding to w 4 = (2,1,1,2): the one correspoinding to T 4 is minimal not require w to be the height function that induces T. In fact, we want to find out what is special about g w,t for T T(A,w). The following lemma describes how g w,t(a,w) looks like compared to the g w,t induced by other polyhedral subdivisions T. Lemma (Crucial Lifting Lemma). Fix w R A and let T be an arbitrary polyhedral subdivision of A. Then the following are equivalent: (i) T T(A,w). (ii) g w,t g w,t for all polyhedral subdivisions T of A. Proof. First, we observe that, by definition of the convex hull, the lifted cell g w,t (σ) lies in the polytope conva w for all polyhedral subdivisions T of A and all σ T. Moreover, T T(A,w) if and only if the lifted cell g w,t (σ) lies in a lower facet of conva w for all σ T. Consider the fiber over x conva w, which is the set P x := { y conva w : y i = x i,i = 1,...,d } This set is a segment, because it is the intersection of the polytope conva w with a line, i.e., an affine subspace. Moreover, the line intersects the lower facets of conva w in the lowest point of P x. This point equals g w,t (x) for all x conva if and only if for all σ T the lifted cell g w,t (σ) lies in a lower facet of conva w. This was shown to be equivalent to T T(A,w). Example (4.1.1 continued). We show in Figures 4.48 to 4.51 the lifting functions on conva 1 for all pairs of heights and triangulations in Example Remember that all given triangulations are regular, and all of them were induced by one of the given heights. In the example it can be seen right away that the lifting according to the triangulation induced by a height is never above any other lifting. We are now in position to prove the following famous and surprising theorem: Theorem (Gelfand, Kapranov, Zelevinsky 1989). C A is the inner normal fan of the secondary polytope Σ(A ). Moreover, the GKZcoordinate construction induces a map φ A from the set of all triangulations of A to the secondary polytope Σ(A ) of A so that φ A (T) is a vertex of Σ(A ) if and only if T is regular. In particular, vertices of Σ(A ) are in oneto-one correspondence with the regular triangulations of A. Furthermore, the face lattice of Σ(A ) is isomorphic to the refinement poset of regular polyhedral subdivisions of A. Proof. Every polytope is determined by its vertices or, equivalently, by its full-dimensional normal cones. The vertices of Σ(A ) must be among the points it is the convex hull of. Therefore, the vertices of the secondary polytope are among the GKZ-vectors of triangulations, and thus it is sufficient to prove the following: for every triangulation T, the normal cone