Doc Math J DMV The Polytope of All Triangulations of a Point Configuration 1 1 Jesus A de Loera Serkan Hosten 2 3 Francisco Santos Bernd Sturmfels Received December Revised March Communicated by G unter M Ziegler Abstract We study the convex hull P of the incidence vectors of A all triangulations of a p oint conguration A This was called the universal p olytop e in The ane span of P is describ ed in terms of the co A circuits of the oriented matroid of A Its intersection with the p ositive orthant is a quasiintegral p olytop e Q whose integral hull equals P We A A present the smallest example where Q and P dier The duality theory A A for regular triangulations in is extended to cover all triangulations We discuss p otential applications to enumeration and optimization problems regarding all triangulations Mathematics Sub ject Classication Primary B Secondary C Introduction We are interested in the set of all triangulations of a conguration A fa a g 1 n d R The subset of regular triangulations is wellunderstoo d thanks to its bijection with the vertices of the secondary polytope see Chapter and Lecture But nonregular triangulations remain a mystery for instance it is still unknown whether any two triangulations of A can b e connected by a sequence of bistellar ips Nonregular triangulations are abundant if A is the vertex set of the cyclic n p olytop e C n there are at least triangulations Prop osition in this 4n4 4 article while the number of regular triangulations is O n 1 Supp orted in part by the National Science Foundation 2 Supp orted in part by the Spanish Direccion General de Investigacion Cientca y Tecnica 3 Supp orted in part by the David and Lucille Packard Foundation and the National Science Foundation Documenta Mathematica De Loera et al One approach in understanding nonregular triangulations is to replace the sec ondary p olytop e by a larger p olytop e P whose vertices are in bijection with all tri A angulations of A The p olytop e P is isomorphic to the universal polytope introduced A by Billera Filliman and Sturmfels They expressed the secondary p olytop e as a n1 pro jection of P and they showed dimP when A is in general p osition A A d+1 We shall now x some notation and dene the p olytop e P A d Throughout this pap er A R will denote a ddimensional conguration of n p ossibly rep eated p oints By a k simplex we mean a subconguration of A consisting of k anely indep endent p oints A triangulation of A is a collection T of d simplices whose convex hulls cover conv A and intersect prop erly for any and in T we have conv conv conv Let A denote the collection of (A) dsimplices in A We dene P as the convex hull in R of the set of incidence A vectors of all triangulations of A For a triangulation T the incidence vector v has T co ordinates v if T and v if T We also consider the T T (A) p olytop e Q a P R which is the linear programming relaxation of A A + P We denote by M A the oriented matroid of ane dep endencies of the p oint A conguration A We rst present linear equations dening the ane hull aP of P These A A equations involve the co circuits see Chapter or Lecture of M A for any + which is a d simplex of A let H b e the hyperplane that contains and let H denote the two op en halfspaces dened by H We recall that the co circuits and H + A H A H of the oriented matroid M A are the resulting partitions A H of A Consider the following linear form X X C o x x + = fag aAH = fag aAH We call C o the cocircuit form asso ciated with the d simplex If conv intconv A we say that is an interior d simplex In this case neither of the two sums in is void Moreover every triangulation T of A contains either no dsimplex containing or exactly two one in the rst sum and one in the second Thus C o vanishes at the incidence vector v of every triangulation of A and hence T on a P We call the equations C o for interior d simplices the interior A cocircuit equations We summarize our main results Theorem Let A be a point conguration with the above conventions (A) i The ane span of P in R is dened by the linear equations C o A for every interior d simplex together with one nonhomogeneous linear equation valid on P A ii P coincides with the integral hul l of Q ie the lattice points in Q are A A A precisely the incidence vectors of triangulations of A iii Two triangulations T and T of A are neighbors in the edge graph of P if and 1 2 A only if they are neighbors in the edge graph of Q A iv For the case of the ngon and congurations with at most d points we have Q P This is not true in general for n d A A Documenta Mathematica The Polytope of All Triangulations The following three types of nonhomogeneous equations may b e used to complete the description of aP in part i of Theorem If is a noninterior d A simplex and conv is a facet of conv A then the co circuit form C o has constant value equal to on P This pro duces new valid equations for a P which we A A call boundary cocircuit equations They can b e expressed in the form X x = fag aAn Another set of valid equations for a P can b e obtained as follows let p A conv A b e a p oint not lying in the convex hull of any d simplex of A Every triangulation of A satises the equation X x (A) pconv ( ) Recall that the chamber complex of A is the common renement of all triangulations of A see We call the equations of type chamber equations b ecause the simplices in the sum only dep end on the chamber in which p lies Note that the b oundary co circuit equations are a particular case of chamber equations d Finally if we denote by v ol the standard volume form on R the following volume equation is satised by every triangulation of A X v ol conv x v ol conv A (A) Remark The interior and b oundary co circuit equations dep end only on the oriented matroid M A of ane dep endencies of A This holds neither for the volume equation nor for chamber equations for example all congurations consisting on the six vertices of a convex planar hexagon have the same oriented matroid while the number of chambers can b e or dep ending on the co ordinates of the vertices Clearly if A has no simplicial facets then there are no b oundary co circuit equa tions However every conguration A has some chamber equations which can b e obtained from the oriented matroid M A Such chambers arise from lexicographic extensions see Figure page Any of them together with the interior co circuit equations will provide a description of a P in terms of M A Part ii A of Theorem implies that this yields a description of P itself in terms of M A A In Section we examine the ane span of P and we prove part i of Theorem A A surprising consequence Corollary is that aP is spanned by the A n1 regular triangulations only This implies the formula dimP when A is A d+1 in general p osition Section contains the pro of of parts ii and iii in Theorem As a consequence of part iii we obtain a combinatorial characterization of the edges of P Theorem Section contains the pro of of part iv We also A discuss computational issues regarding the enumeration of triangulations and the optimization of linear cost functions over P In Section we present a duality A theory relating nonregular triangulations of A with virtual chambers in the Gale transform of A Documenta Mathematica De Loera et al Equations defining the affine span of P A We introduce now some basic denitions and prop erties concerning regular triangu lations For a more detailed description and the relevant background the reader may consult Chapter and A regular triangulation of A is a triangulation which is obtained by pro jecting the lower envelope of a d dimensional simplicial p olytop e onto conv A In other words a triangulation is regular if it supp orts a piecewise convex linear functional In Chapter the collection of regular triangulations of a p oint conguration is identied with the vertex set of a p olytop e A of dimension n d embedded n in R This p olytop e called the secondary polytope is a pro jection of P The A P (A) A e where e and pro jection map R R is given by e v ol a a e denote the standard basis vectors a A characterization of the edges of A is given in Chapter This uses the notions of circuits and bistel lar ips We only dene bistellar ips in the general p osition case a circuit of the p oint conguration A is a minimal anely dep endent set If A is in general p osition circuits are subsets of cardinality d The unique up to scaling factor ane dep endency equation satised by a circuit Z splits it into + two subsets Z and Z consisting of the p oints which have p ositive and negative + co ecients resp ectively Any circuit Z has exactly two triangulations tZ fZ n + fag a Z g and tZ fZ n fag a Z g If a triangulation T of A contains + + one of the two triangulations of a circuit Z say tZ then T T n tZ tZ is again a triangulation of A The op eration that passes from T to T or vice versa is called a bistel lar ip Two regular triangulations are neighbors in the skeleton of the secondary p olytop e A if and only if they dier by a bistellar ip This implies that any two regular triangulations can b e transformed to one another by a nite sequence of bistellar ips It is unknown whether this prop erty is true for nonregular triangulations Our next goal is to prove part i of Theorem We rst state a lemma ab out the b ehavior of triangulations under