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

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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

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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

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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 the matroidal op erations of deletion A

Ana and contraction A Aa where a is a vertex of A We regard Aa as a

i i i i

conguration of typically n p oints maybe less if a was a rep eated p oint in ane

i

d space The convex hull of Aa is the vertex gure of conv A at a

i i

Lemma Let a A be a vertex of conv A Every triangulation of Ana can

1 1

be extended to a triangulation of A Every regular triangulation of Aa can be

1

extended to a triangulation of A The latter fails for nonregular triangulations

Proof Every regular or nonregular triangulation T of Ana can b e extended to a

1

triangulation T of A by the placing op eration describ ed on page in In this

situation T is regular if and only if T is regular

The extension prop erty of regular triangulations for contractions follows from

the identication of the secondary fan of A with the chamber complex of B where B

is a Gale transform of A see in particular Lemma and the paragraph after

Lemma The reasoning is this let T b e a regular triangulation of Aa It

1

of B nb where b B is the p oint corresp onding to corresp onds to a chamber C

1 1 T

a in the Gale transform The chamber C may split into smaller chambers when

1 T

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The Polytope of All Triangulations

passing to the chamber complex of B and any such smaller chamber C corresp onds

T

to a triangulation T of A with T a T

1

3

To see that regularity is necessary in the previous paragraph let A R b e

the conguration a a a a

1 2 3 4

a a and a Let T b e the triangulation

5 6 7

of Aa given by the triangles f g f g f g f g f g f g

7

and f g There is no triangulation T of A such that T a T The nonconvex

7

p olyhedron with vertices a a and triangular faces determined by the ab ove list

1 6

together with the triangle f g is the Schonhardt polyhedron which cannot b e

triangulated without adding a new p oint

Part i of Theorem is a consequence of the following theorem and the exis

tence of nonhomogeneous forms vanishing on P

A

P

c x c R be any homogeneous linear form on Theorem Let h

(A)

(A)

R The fol lowing properties are equivalent

i h is a linear combination of the interior cocircuit forms C o

ii h vanishes on the incidence vector of every triangulation of A

iii h vanishes on the incidence vector of every regular triangulation of A

Proof iiiiii are obvious We prove iiii

r eg

denote the convex hull of all p oints v where T is a regular triangulation Let P

T

A

P

r eg r eg

P Let h c x b e any linear form which vanishes on P of A Thus P

A

A A

We shall prove that h is a linear combination of the interior co circuit forms using a

double induction on n jAj and d dimA Assume that the statement is true for

any conguration of smaller cardinality or smaller dimension

d

Let a b e a vertex of conv A Let us supp ose that Ana still spans R Oth

1 1

r eg r eg

and P erwise P are anely isomorphic and the theorem follows by induction

A

Ana

1

r eg

If a then C o involves at most The interior co circuit forms C o vanish on P

1

A

one dsimplex of the form fa g Subtracting appropriate multiples of those

1

C o from h we get another linear form h in which the variables x corresp onding to

1

these simplices do not app ear That is

X X

h c x c x

1

:a

1

:a

1

conv ( na ) boundar y (conv (A))

1

P

r eg

c x is a linear form vanishing on P Indeed let The second sum h

2

:a

Ana

1

1

T b e any regular triangulation of Ana Pick a regular triangulation T of A that

1

extends T as in Lemma Since a is a vertex of conv A the triangulation T

1

cannot contain a simplex of the form fa g where conv is in the b oundary of

1

conv A This fact together with h v implies h v and consequently

1 T 2 T

h v

2 T

Every co circuit form C o of Ana is either a co circuit form of A as well or can

1

b e extended to a co circuit form of A by adding a single variable x with the

fa g

1

appropriate sign By induction hypothesis h is a linear combination of the co circuit

2

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forms of Ana We extend this presentation to a linear combination of co circuit forms

1

r eg

of A which vanishes on P We subtract it from h to get a new form h which

1 3

A

r eg

vanishes on P and involves only dsimplices of the form a

1

A

X

h c x

3

:a

1

The assignment fa g denes a bijection b etween the simplices of Aa and the

1 1

r eg

simplices of A containing a Therefore we can interpret h as a linear form on P

1 3

Aa

1

r eg

Lemma guarantees that h vanishes on P By the induction hypothesis h

3 3

Aa

1

is a linear combination of co circuit forms C o of Aa We replace each variable x

1

in this linear combination by the corresp onding variable x This transforms

fa g

1

co circuit forms of Aa into co circuit forms C o of A Therefore h is a linear

1 3

fa g

1

combination of co circuit forms of A This proves Theorem and Theorem i

(A)

Corollary The linear subspace of R parallel to aP is spanned by al l

A

where T and T are regular triangulations of A diering by a bistel lar vectors v v

T T

ip

Proof It follows from Theorem that the linear subspace in question is spanned

where T and T are regular triangulations Since every pair by all vectors v v

T T

of regular triangulations is connected by a sequence of bistellar ips the corollary

follows

Theorem implies dimP jAj R where R is the rank of the

A

n

interior co circuit forms If the p oints of A are in general p osition then jAj

d+1

where T and T are regular triangulations of A In this case the vector v v

T T

+

+

diering by a bistellar ip equals v v where tZ and tZ are the two

t(Z ) t(Z )

triangulations of the circuit Z on which the bistellar ip is supp orted

d

Theorem Let A R be a conguration of n points in general position Let

a A and R be the rank of the interior cocircuit forms

1

i The interior and boundary cocircuit forms C o for which a form a basis

1

for the space of linear forms vanishing on the linear space parallel to P Thus

A

n1

R

d

+

ii The vectors v v for the circuits Z containing a form a basis of the

1

t(Z ) t(Z )

n1

linear space parallel to P Thus dimP

A A

d+1

Proof The co circuit forms C o for the d simplices not containing a are

1

linearly indep endent b ecause each simplex containing a app ears in exactly one

1

n1

+

of them Thus R Likewise the vectors v v for the circuits

t(Z ) t(Z )

d

Z containing a are linearly indep endent b ecause each simplex not containing a

1 1

n1

app ears in exactly one of them Thus dimP This together with the

A

d+1

n

formula dimP R nishes the pro of

A

d+1

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The Polytope of All Triangulations

Proposition If A is a conguration in general position then aP is dened

A

by the chamber equations

Prop osition is generally false for congurations in sp ecial p osition b ecause

a collection of dsimplices may uniquely cover all op en chambers without b eing a

triangulation For example the conguration consisting of the four vertices of a

quadrilateral plus the intersection of its diagonals has triangulations dimP

A

while the chamber equations have nonnegative integer solutions whose convex hull

is a dimensional p olytop e For the vertex set of the cub e we have calculated that

dimP but the chamber equations dene an ane space of dimension

A

Prop osition is implied by Theorem and the following lemma which expresses

interior co circuit forms as dierences of chambers

d

Lemma Let A be a conguration in general position in R Let C and C be

1 2

two neighboring maximal chambers and the unique d simplex containing their

common facet Then

X X

x C o x

:C conv ( ) :C conv ( )

2 1

+

Proof Let H b e the hyperplane dened by with halfspaces H C and H

1

C If is any dsimplex which contains C then either contains C as well or

2 1 2

+

a where a H similarly for C

2

If A is in sp ecial p osition then more than one d simplex may contain the

common facet of C and C Call the collection of them In this case with similar

1 2

arguments one can prove that the formula in Lemma has to b e corrected by

P

C o for C o substituting

Remark Let M b e the incidence matrix of the chambers and the dsimplices of

A If A is in general p osition then Prop osition implies that

(A)

aP fx R M x 1g

A

Row and column bases of M have b een studied in and and a formula is given

for rankM in The formulae in Theorem are a sp ecial case of that formula

The relation between P and Q

A A

Part i of Theorem implies that the linear programming relaxation Q of P is

A A

dened by the interior co circuit equations C o plus an extra nonhomogeneous

equation satised on P and the inequalities x for each simplex of A Clearly

A

P Q We shall examine the relationship b etween these two p olytop es

A A

(A)

We call support of a p oint v R and denote it suppv the collection of

dsimplices for which v

(A)

Lemma i Q is a subpolytope of the unit cube conv x x f g

A

ii Every vertex of P is also a vertex of Q

A A

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iii If v is any point in Q then suppv covers conv A ie

A

conv conv A

: supp(v )

iv A vertex v of Q is a vertex of P if and only if its support contains a trian

A A

gulation

Proof The chamber equations are valid for Q and they imply part iii Also

A

since every x app ears in at least one of them the nonnegativity constraints imply

x for all This proves part i which in turn implies part ii The only

ifdirection of iv is obvious For the ifdirection supp ose rst that suppv is the

supp ort of a triangulation T Then the chamber equations imply that v is the inci

dence vector of T hence is a vertex of P If suppv strictly contains a triangulation

A

v v

T

T then it cannot b e a vertex of Q b ecause v is still a p oint in Q for a

A A

1

suciently small p ositive But then v v v is not a vertex of Q

T A

Theorem Every integral point of Q is the incidence vector of a triangulation

A

of A ie P is the integral hul l of Q

A A

Proof Let v b e an integral p oint of Q By Lemma iii we only need to prove

A

that any two simplices in suppv intersect prop erly Supp ose this is not the case for

two simplices and in suppv ie

1 2

conv conv conv

1 2 1 2

Take a p oint a in conv conv nconv Then the minimal face

1 2 1 2

subset F of with a conv F is not a face of For each simplex of suppv

1 2

having F as a face consider the convex p olyhedral cone

c a posconv a f p a p conv g

Note that the facets of c are in to corresp ondence with the facets of which

contain F We claim that conv A is contained in the union of such cones Supp ose

a p oint b of conv A lies outside their union Then b sees a facet of some cone

c where suppv Let b e the corresp onding facet of which contains F

By the choice of there is no dsimplex in suppv having as a facet and lying in

the halfspace containing b This violates the interior co circuit equation C o v

since v Therefore an op en neighborho o d of a in conv A is covered by those

simplices in suppv which have F as a face The interior of one of these simplices

intersects the interior of conv This violates the chamber equations for v

2

We next prove that the edges of P are also edges of Q A dierent pro of for

A A

this theorem can b e derived from more general results ab out p olytop es due to

Matsui and Tamura Here we present a selfcontained pro of in the context of

triangulations

Theorem Let T and T be two distinct triangulations of A The fol lowing state

1 2

ments are equivalent

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The Polytope of All Triangulations

are not neighbors in Q and v i v

A T T

2 1

are not neighbors in P and v ii v

A T T

2 1

iii There exist two triangulations T and T of A dierent from T and T such

3 4 1 2

v v v that v

T T T T

4 3 2 1

iv There exist partitions T R L and T R L such that L R and

1 1 1 2 2 2 1 2

R L are two other triangulations of A dierent from T and T

1 2 1 2

Proof iviiiiii are obvious We only need to show iiv We dene a

graph G whose no des are the dsimplices of T Two dsimplices of T are adjacent in

1 1

G if and only if they share a common facet and this facet is not a facet of any dsimplex

of T The graph G has the following prop erty If H is a connected comp onent of

2

G then the top ological b oundary of conv is the union of d simplices

H

which are faces of b oth T and T

1 2

Next we will construct L L R and R Let b e a simplex of T which is not

1 2 1 2 0 1

in T Let L b e the collection of dsimplices in the same connected comp onent of G as

2 1

conv conv and jLj and let R T nL Moreover let jR j

L 0 1 1 1 R

1 1

Finally let L resp R b e the collection of dsimplices of T whose convex hull

2 2 2

intersect the interior of jLj resp of jR j By the prop erty of G mentioned ab ove no

simplex of T can intersect the interiors of b oth jLj and jR j Thus T is the disjoint

2 2

union of R and L Also by the same prop erty the simplices of L R same

2 2 1 2

for R L intersect prop erly Moreover they cover conv A b ecause their union

1 2

covers conv A twice We conclude that the disjoint unions L R and L R

1 2 2 1

are triangulations of A Clearly L L b ecause L nL Let us assume that

1 2 0 1 2

are neighbors in Q This will nish the and v R R and prove that then v

A T 1 2 T

2 1

pro of of the theorem

This face and v Let v b e a p oint in the minimal face F of Q containing v

T A T

2 1

equal to zero Thus or v is dened setting all co ordinates not app earing in v

T T

2 1

suppv T T The entry of v corresp onding to any dsimplex in R R equals

1 2 1 2

b ecause of the chamber equations On the other hand for any two dsimplices

1

since these v and adjacent in G the interior co circuit equations imply v

2

2 1

two simplices are the only ones in T T having as a facet Thus the

1 2 1 2

entries of v corresp onding to simplices in L have a constant value With this the

1

chamber equations imply that the entries corresp onding to simplices in L have a

2

This implies that F is a segment v constant value Thus v v

T T

2 1

are neighbors and v ie that v

T T

2 1

The ab ove theorem implies that any two integral vertices of Q triangulations

A

are connected by a path of integral vertices Note that it is still conceivable that we

could have two triangulations of A which are not connected by bistellar ips

Examples and applications

Our linear programming relaxation Q is generally not a lattice p olytop e Therefore

A

P is strictly contained in Q In this section we will exhibit some cases where

A A

P Q and the smallest conguration for which P Q

A A A A

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d

Theorem Let A R be a conguration of n points The equality P Q

A A

holds in the fol lowing cases

i d and al l points lie on the boundary of a convex polygon

ii d

iii n d

Proof i Let v b e a vertex of Q Let S b e a subset of suppv where all triangles

A

in S intersect prop erly and cover a convex subp olygon of conv A Supp ose that S

is maximal with these two prop erties Let e b e an edge of the subp olygon covered

by S Then S is contained in one of the two halfplanes dened by e By maximality

of S and the interior co circuit equations e must b e a segment on the b oundary of

conv A This proves that S covers conv A and hence is a triangulation Lemma

iv implies that v is a vertex of P

A

ii The pro of is a minor variation of case i

iii Let S fT T g b e the collection of all triangulations of A By Corollary

1 k

b elow every triangulation T of A contains a simplex which is not contained in

i i

any other triangulation Therefore setting the co ordinate of equal to zero denes

i

a facet of P that contains every triangulation but T Thus P is a k simplex

A i A

The fact that all facets of P are dened by setting co ordinates equal to zero implies

A

that P Q

A A

Example A fractional vertex of Q

A

For any A with P Q we must have n d A minimal example is provided

A A

by the vertices of a regular p entagon plus its center This conguration is

in general p osition and has triangles and triangulations Consider the vector

20

v R with co ordinates v v v v v v

f123g f234g f345g f145g f125g f013g

v v v v and all other co ordinates zero It satises

f024g f035g f014g f025g

the interior and b oundary co circuit equations Therefore v lies in Q Since suppv

A

do es not contain any triangulation P Q This fractional p oint is the only vertex

A A

of Q which is not in P

A A

Remark The prop erty P Q is neither sucient nor necessary for a cong

A A

uration A to have all triangulations regular In Example all triangulations are

regular but P Q For the canonical example of the planar conguration which has

A A

nonregular triangulations six p oints which form two triangles with parallel edges

we still have P Q

A A

Let C b e the vertex set of a planar ngon The following prop osition gives an

n

n1

Q irredundant inequality presentation of the dimensional p olytop e P

C C

n n

3

are dened by x where is a Proposition For n the facets of P

C

n

triangle with at most one of its edges lying on the boundary of C

n

Proof We call a triangle external if it has two edges on the b oundary of C We rst

n

Without show that for any external triangle x do es not dene a facet of P

C

n

loss of generality we can assume f ng Supp ose that x denes a facet of

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The Polytope of All Triangulations

n1

Then P anely indep endent triangulations not involving together with a

C

n

3

The triangulation T con triangulation T that contains will anely span aP

C

n

tains a triangle fi ng for some i Since n there exists another triangulation

S which contains fj ng where j i But then S cannot b e expressed as an ane

combination of the ab ove set of triangulations since fj ng cannot app ear in any of

them This contradiction shows that an external triangle do es not dene a facet of

Q

C

n

For n a direct investigation shows that the ve nonexternal triangles corre

sp ond to facets For n supp ose is a nonexternal triangle Then there exists a

vertex i such that C C ni contains as a nonexternal triangle Without loss of

n1 n

generality we can assume i Let b e the external triangle f ng By induction

n2

x denes a facet of C In other words there are anely indep endent

n1

3

triangulations of C which do not contain This set of anely indep endent tri

n1

angulations can b e extended to triangulations of C by adding Now we need to

n

n2

pro duce additional anely indep endent triangulations which do not contain

2

for each j and k such that j k n we can construct a triangulation which

contains the triangles f j g f j k g and f k ng and hence not and which

do es not contain Similarly for each l n we can construct triangulations

n2

which contain f l g and f l ng These additional triangulations together

2

n1

with the previous ones form a set of none anely indep endent p oints in P

C

n

3

of them containing

whenever Remark The argument that x do es not dene a facet of Q

C

n

is an external triangle can b e generalized The external simplices of a p oint

conguration A in general p osition are the following if any vertex a of convA is

i

deleted then the p oint conguration Ana will again b e in general p osition Each

i

facet of conv Ana that is visible from a together with a will form a simplex

i i i

which will not dene a facet for Q The argument is identical to the one ab ove

A

We close this section with remarks ab out using Q to enumerate the triangula

A

tions of A and to solve optimization problems over P If A is in general p osition

A

then by Remark

(A)

Q fx R M x 1g

A

+

(A)

P conv fx f g M x 1g

A

This means that P is a set partitioning polytope Even if A is not in general p osition

A

by introducing extra variables for the interior d simplices of A P can b e realized

A

as a set partitioning p olytop e This has imp ortant implications for enumeration and

optimization purp oses For example Trubin shows that if P is a set partitioning

p olytop e and Q its linear relaxation then P is quasiintegral ie every edge of P is

also an edge of Q Balas and Padberg and Matsui and Tamura have given

a characterization of adjacency b etween vertices of P This leads to an algorithm

for optimizing linear functionals over P using Q Starting from an integral vertex of

Q the algorithm nds the optimal solution visiting only integral vertices of Q in a

fashion similar to the simplex metho d in linear programming The same adjacency

characterization can b e used to enumerate the vertices of P as well

Unfortunately no implementation of the BalasPadberg pro cedure is known to us

Moreover enumerating the triangulations of A using the existing vertex enumeration

Documenta Mathematica

De Loera et al

packages has two ma jor drawbacks First of all an inequality presentation of P is

A

either hard to get or it might involve to o many constraints Secondly if one uses Q

A

instead of P then there is an excessive number of fractional vertices which have to b e

A

enumerated to o Our exp eriments using several vertex enumeration packages allowed

us only to compute small examples However the situation is more promising for

optimization problems over P thanks to the ecient implementations of branchand

A

bound and branchandcut algorithms for integer programming Finding triangulations

using minimum or maximum number of simplices or nding a minimummaximum

cost triangulation fall into the category of these optimization problems In order to

illustrate the sizes of the problems one can attempt to solve and the eciency of using

linear and integer programming techniques over P we give the following example

A

Triangulating the ddimensional cub e with minimal number of simplices is imp ortant

for their use in algorithms for the computation of xed p oints of continuous maps

P

This is equivalent to minimizing the functional x over P where A is the vertex

A

set of the dcub e In the case of the cub e d n the system dening Q has

A

equations and variables Using CPLEX on a SPARC workstation it

takes seconds to verify that the cub es minimal triangulation has simplices

The size of the corresp onding linear programs for the dcub e with d gets to o

big In this case one should formulate a smaller linear program which exploits the

symmetries of the dcub e This was done successfully for d in

Duality

In this section we extend the duality theory in to cover all triangulations Two

applications are included a short pro of of Carl Lees result that every triangulation

of a conguration of d p oints is regular and an exp onential lower b ound for

the number of triangulations of the cyclic p olytop e C n

4n4

Following we now consider a conguration A fa a a g of vectors

1 2 n

d+1

n

spanning R Thus jAj and equality holds if A is in general p osition If

d+1

there exists an ane hyperplane H H which intersects posa for every a A

i i

then we can consider A as a ddimensional p oint conguration in H identied with

d

R If this is the case A is said to b e acyclic and we are in the setting of the

previous sections For the nonacyclic case see Remark b elow

nd1

Let B fb b b g b e a spanning subset of R which is a Gale trans

1 2 n

P

d+1 nd1

n

form of A This means that a b in R R In particular

i i

i=1

M B is the oriented matroid dual to M A

(A)

Let T A f g b e the set of incidence vectors of all triangulations of A

and T A the subset corresp onding to regular triangulations Similarly let A

r eg

(A)

f g b e the set of all chambers of A By the results in

T A B and T B A

r eg r eg

(A)

We identify each co circuit form C o of A with its co ecient vector in f g

Let C oA denote the collection of all cocircuit vectors C o where runs over all

linearly indep endent dsubsets of A and let C o A b e the subset of interior co circuit

int

vectors Recall that C o is interior if and only if b oth and app ear among

the co ordinates of C o Dually let b e any spanning d subset of A Then

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The Polytope of All Triangulations

contains a unique signed circuit Z Z Z of A We dene the circuit vector

+

X X

e e C i

nj ni

j Z iZ

+

(A)

The e are standard basis vectors in R We say that C i is an interior circuit

ni

vector if Z and Z Let C iA denote the set of all circuit vectors and

+

C i A the subset of interior circuit vectors A is acyclic if and only if C iA

int

(A)

C i A We x the standard inner pro duct h i on R Here is our rst

int

duality theorem

d+1 nd1

Theorem Let A R and B R be Gale transforms of each other

i Circuit and cocircuit vectors satisfy C iA C oB and C iB C oA

ii If A is in general position then the subspaces spanned by C iA and C oA are

(A)

orthogonal complements in R

Proof Recall the following two facts from oriented matroid duality A d

subset of A is spanning if and only if the complementary n d subset of B is

linearly indep endent The signed circuits of A are the signed co circuits of B and vice

versa These two facts imply assertion i

ii Let C i b e a circuit vector and let C o b e a co circuit vector If suppC i

suppC o then hC o C i i Otherwise suppC i suppC o f g

1 2

where f ig nj and f j g ni If a and a are in dierent halfspaces

1 2 i j

dened by then f ig and f j g have the same sign in C i since they app ear in the

same triangulation of If a and a are in the same halfspace then f ig and f j g

i j

have opp osite signs in C i since they do not b elong to the same triangulation of

This shows

hC o C i i for all C o C oA C i C iA

n1

If A is acyclic by the pro of of Theorem we have dimC oA and

d

n1

dimC iA We conclude that C iA and C oA span orthogonal comple

d+1

ments If A is in general p osition but not acyclic then B is acyclic and the result

follows from i

The orthogonality relation need not hold for congurations A in sp ecial p o

sition For example let A fa a a g b e the vertex set of the regular o ctahe

1 2 6

dron where a and a are not connected by an edge If we choose f g and

5 6

f g then hC o C i i

Remark The main results of Section and can b e extended to vector cong

urations The dimension formula given for p oint congurations in general p osition

n1

should b e corrected to dimP whenever A is a nonacyclic conguration

A

d+1

in general p osition This can b e proved using duality

Proposition For every vector conguration A the orthogonal complement of

C iA is contained in C oA and vice versa

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De Loera et al

Prop osition can b e deduced from the following theorem

(A)

Theorem For a vector X f g the fol lowing are equivalent

a T T A hX T i and C i C i A hX C i i

r eg int

b T T A hX T i and C i C i A hX C i i

int

c X is the incidence vector of a triangulation of B that is the set of al l n d

subsets satisfying X denes a triangulation of B

f1ngn

Proof The equivalence of a and b follows from aT A aT A which is

r eg

a consequence of Corollary Using Theorem i and equation we see that

a is equivalent to

C B hX C i and C o C o B hX C o i

int

The solutions X to this are the triangulations of B by Theorem

(A)

Corollary If A is in general position then for a vector X f g the

fol lowing are equivalent

a hX T i for al l T T A

r eg

b hX T i for al l T T A

c X is the incidence vector of a triangulation of B

The chambers of A constitute a generally prop er subset of the vectors X

(A)

f g characterized by the three equivalent conditions in Theorem We pro

p ose the following interpretation for the remaining solutions

(A)

Definition The solutions X f g to the system a in Theorem

viewed as collections of d subsets in A are called the virtual chambers of A

Writing for the set of virtual chambers we have

v ir t

T A B and T B A

v ir t v ir t

Remark There are two kinds of virtual chambers in B n B the rst

v ir t

kind of these can b ecome real chambers in a dierent realization of M B These

corresp ond to the triangulations of A which are regular in some other realization

of M A The second kind the truly virtual chambers will never show up as real

chambers and thus they corresp ond to nonregular triangulations which never b ecome

regular An example of the second kind can b e found in Prop osition in

We now present two applications of our duality results

d+1

Proposition Carl Lee If A is a vector conguration in R with jAj

d then every triangulation of A is regular

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The Polytope of All Triangulations

8

9 7

10 6

11 5

12 4

1 3

2

Figure A Gale diagram of C

12

Proof If jAj d there is a trivial triangulation which is regular The case

jAj d is still very easy since the linear transform B of A is onedimensional Let

us assume that jAj d Then B is twodimensional and its simplices are pairs of

indep endent vectors Let C b e a virtual chamber of B and let fb b g b e a simplex

1 2

in the supp ort of C We have the following

a if b B is such that posb b posb b then fb b g is in the supp ort of C

1 2 1 1

b if b B lies in posb b then exactly one of fb b g and fb b g is in the supp ort

1 2 1 2

of C

Both prop erties follow from C i C i B hC C i i Moreover since

int

T T B hC T i a simplex fb b g where posb b posb b cannot b e in

i j i j 1 2

C From these we conclude that there is a unique minimal simplex lying in C This

implies that C is a real chamber

d+1

Corollary If A is a vector conguration in R with jAj d then every

triangulation of A contains a simplex which is not used in any other triangulation of

A

Proof For jAj d and for jAj d the statement is again trivial For the

case jAj d the minimal simplex of C in the pro of of the Theorem is such a

simplex

The triangulations of all cyclic p olytop es C n are known to b e connected by

d

bistellar ips We close the pap er with a result ab out the abundance of non

regular triangulations of C n

d

4

Proposition If A is the vertex set of C n then A has O n regular

4n4

n

triangulations and has at least triangulations

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De Loera et al

8 8 9 7 9 7

10 6 10 6

11 5 11 L 5

4 4 12 12

1 3 1 3

2 2

Figure A p erturbation that creates a chamber

4

Proof The fact that A has O n regular triangulations was established in Theo

rem note that their d corresp onds to the d in our notation A Gale

transform B fb b b g of A can b e depicted as a ngon under tak

1 2 4n

ing the antipo dals of fb b b g see Figure and also compare with Figure

2 4 4n

in That conguration can b e assumed to b e a regular ngon in particu

lar the subconguration fb b b g is a regular ngon Hence the cones

1 3 4n1

posb b i n and i o dd intersect in a halfline L Now by p erturbing

i i+2n

n

the vertices of this ngon we can create dierent chambers which corresp ond to

n

distinct triangulations of A see Figure These chambers are virtual chambers

of A b ecause the small p erturbation do es not change the oriented matroid This

n

shows that C n has at least nonregular triangulations All of them b ecome

4n4

regular for some p oint conguration combinatorially equivalent to A

Acknowledgments We thank Victor Reiner for helpful remarks and questions We

are also grateful to David Avis for his help on using his vertex enumeration package

RS

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