On the integer points in a lattice polytope: n-fold Minkowski sum and boundary
aa r X i v : . [ m a t h . M G ] J un On the integer points in a lattice polytope: n -fold Minkowski sum and boundary Marko Lindner and
Steffen Roch
November 15, 2018
Abstract.
In this article we compare the set of integer points in the homothetic copy n Π of alattice polytope Π ⊆ R d with the set of all sums x + · · · + x n with x , ..., x n ∈ Π ∩ Z d and n ∈ N .We give conditions on the polytope Π under which these two sets coincide and we discuss two notionsof boundary for subsets of Z d or, more generally, subsets of a finitely generated discrete group. Mathematics subject classification (2000):
Keywords and phrases: lattice polytopes, integer points, boundary, projection methods
Throughout, we denote by N , Z , Q and R the natural, integer, rational and real numbers,respectively, and we fix a number d ∈ N . For arbitrary n ∈ N , we compare the set of all integerpoints in the homothetic copy n Π of a lattice polytope Π ⊆ R d (that means Π is the convex hullof a finite number of points in Z d ) with the set of sums x + · · · + x n with x , . . . , x n ∈ Π ∩ Z d .It is easy to see that the latter set is always contained in the first – but in general, they aredifferent. We give conditions on the polytope under which these two sets coincide and wediscuss two notions of boundary for subsets of Z d and, more generally, of a finitely generated(not necessarily commutative) discrete group.The motivation for this paper stems from the study of projection methods for the approximatesolution of operator equations. Let A be a bounded linear operator acting on a Banach space.To solve the operator equation Au = f numerically, one chooses a sequence ( Q n ) of projections(usually assumed to be of finite rank and to converge strongly to the identity operator) andreplaces the equation Au = f by the sequence of the linear systems Q n AQ n u n = Q n f . Whatone expects is that, under suitable conditions, the solutions u n of these systems converge to thesolution u of the original equation. If the Banach space on which A lives consists of functions ona countable set Y (think of a sequence space l ( Y ), for example) then it is convenient to choosean increasing sequence ( Y n ) of finite subsets of Y and to specify Q n as the operator P Y n whichrestricts a function on Y to Y n .In this paper, we will be concerned with the case when Y is a finitely generated discrete groupΓ. In case Γ is the additive group Z d , a typical (rather geometric) approach [15, 17, 13, 14] todesign a projection method is to fix a compact set (for example a lattice polytope) Π ⊆ R d andto consider the operator P Π n of restriction to (likewise, the operator of multiplication by thecharacteristic function of) the set Π n := ( n Π) ∩ Z d (1)1or n ∈ N . If we assume that the origin is in the interior of Π then it follows thatΠ n ⊆ Π n +1 for all n ∈ N and [ n ∈ N Π n = Z d , (2)whence the sequence (Π n ) n ∈ N gives rise to an increasing sequence of finite-dimensional projectionoperators on l ( Z d ) that strongly converges to the identity operator as n → ∞ .In the case of a general finitely generated group Γ, this geometric approach is clearly infeasible.A natural idea here is to fix a finite set Ω ⊆ Γ of generators of Γ (that is, we assume that Ωgenerates Γ as a semi-group) and to consider the setΩ n := { x x · · · x n : x , x , . . . , x n ∈ Ω } (3)of all words of length n ≥ x − ∈ Ω if x ∈ Ω), contains the identity element e of Γ (in analogy to the above approachin Z d ) and if Γ is equipped with the word metric over Ω then Ω n is the disk of radius n in Γaround the identity e . Moreover, one gets that also (Ω n ) n ∈ N yields an increasing sequence offinite-dimensional projections with strong limit identity, i.e., (2) holds with Π m replaced by Ω m and Z d by Γ.A natural question before proposing the latter approach for general finitely generated groupsΓ is whether or not the geometric approach (1) and the algebraic approach (3) coincide if wehave Γ = Z d and use Ω := Π ∩ Z d as finite set of generators in (3) with a symmetric latticepolytope Π ⊆ R d containing the origin in its interior. This question is discussed in Section 2.We will give conditions on the polytope Π under which (1) and (3) coincide – but in generalthey do not.In Section 3 we address a further question that arises in the study of projection methods. In[19] it has been pointed out that the “boundaries” ∂ Ω Ω n (if appropriately defined) of the setsΩ n (or Π n ) hold the key to the answer to whether or not the projection method P Ω n AP Ω n u n = P Ω n f, n ∈ N , yields stable approximations u n to the solution u of Au = f . In Section 3 we give special empha-sis to the question whether the “algebraic boundaries” ∂ Ω Ω n coincide with the corresponding“numerical boundaries” Ω n \ Ω n − . Now fix d ∈ N , let e , ..., e d be the standard unit vectors of R d , and denote the unit simplexconv { , e , ..., e d } by σ d . (For standard notions on convex polytopes we recommend [9, 8, 22];for lattice polytopes see [3, 6, 7, 20, 21].)Given a non-empty subset S of R d and a positive integer n , we write nS := { ns : s ∈ S } and n ∗ S := { s + · · · + s n : s , ..., s n ∈ S } = S + · · · + S for the ratio- n homothetic copy and n -fold Minkowski sum of S , respectively. For convenience,we also set 0 S := { } and 0 ∗ S := { } . It is easy to see that nS = n ∗ S holds for all n ∈ N if S is convex. Indeed, the inclusion nS ⊆ n ∗ S is always true, and, by convexity of S , s + · · · + s n ns with s = ( s + · · · + s n ) /n ∈ S for all n ∈ N and s , ..., s n ∈ S . (Note thatequality of nS and n ∗ S for all n ∈ N does not imply convexity of S , as S = Q shows.)For a convex set Π ⊆ R d containing at least two integer points, it is clear that ( n Π) ∩ Z d = n (Π ∩ Z d ) as soon as n >
1. But (as motivated in the introduction) a much more interestingquestion is whether or not ( n Π) ∩ Z d = n ∗ (Π ∩ Z d ) (4)is true for all n ∈ N . We will study this question for certain polytopes Π.Let v , ..., v k be points of Z d with their affine hull equal to R d and put Π = conv { v , ..., v k } .Π is a so-called lattice polytope as all its vertices are in Z d . We will suppose that there is noproper subset I of { , ..., k } with Π = conv { v i : i ∈ I } , so that v , ..., v k are the vertices of Π. IfΠ ∩ Z d only consists of the vertices of Π then Π is called an elementary polytope [10, 20] (or a lattice-(point-)free polytope [2, 11]). The following lemma is fairly standard: Lemma 2.1
Let A ∈ Z d × d be a matrix and a , ..., a d ∈ Z d its columns. a) The following conditions are equivalent:(i) A ( Z d ) = Z d .(ii) A − is an integer matrix.(iii) det A = ± .(iv) The parallelotope A ([0 , d ) spanned by a , ..., a d has volume .(v) The parallelotope A ([0 , d ) is elementary. b) The condition(vi) The simplex A ( σ d ) = conv { , a , ..., a d } is elementary.is necessary for ( i ) – ( v ) ; it is moreover sufficient iff d ∈ { , } . If ( i )–( v ) hold then A is called an integer unimodular matrix [7, 20] and the simplex A ( σ d )has volume 1 /d ! and is sometimes called a primitive (or unimodular ) simplex (e.g. [10]). Soprimitive simplices are elementary, and the converse holds iff d ∈ { , } .For the sake of completeness, we give a short sketch of the proof of Lemma 2.1: Proof.
Part a) follows by standard arguments using det A − = 1 / det A and vol( A ([0 , d )) = | det A | vol([0 , d ). The implication ( v ) ⇒ ( vi ) holds by σ d ⊆ [0 , d . For d = 1, the implication( vi ) ⇒ ( v ) is clear by σ = [0 , . For d = 2, if x is an integer non-vertex point in A ([0 , ),then also a + a − x is an integer non-vertex point in A ([0 , ). But one of the two points isin A ( σ ), so that ( vi ) ⇒ ( v ) holds. For d ≥
3, there are elementary but not primitive simplices(see Examples 2.2 a and b below).Here are two slightly different constructions leading to elementary but not primitive simplicesin dimension d ≥ Example 2.2 a)
Let d ≥
3, fix an m ∈ N , take a := e , a := e , ... , a d − := e d − ∈ Z d and a d := ( − , ..., − , m ) ⊤ , and let A ∈ Z d × d be the matrix with columns a , ..., a d . ThenΣ d,m := A ( σ d ) = conv { , a , ..., a d } is primitive iff m = det A = 1. On the other hand, the number of integer points in Σ d,m apart from its d + 1 vertices is equal to k = ⌊ m/d ⌋ (integer division), and these k other integerpoints are 1 e d , ..., ke d . (To see this, look at the projection of Σ d,m to the hyperplane spannedby e , ..., e d − .) So for m ∈ { , ..., d − } we have an elementary but not primitive simplex.3 ) If we change the last column in the above example from a d = ( − , ..., − , m ) ⊤ to a ′ d :=(1 , ..., , m ) ⊤ with m ∈ N and call the new matrix A ′ then, again, the simplexΣ ′ d,m := A ′ ( σ d ) = conv { , a , ..., a d − , a ′ d } is not primitive for m = det A ′ ≥ m ∈ N . So this simplexΣ ′ d,m can have arbitrarily large volume m/d ! without containing any integer points other thanits vertices. The simplices Σ ′ d,m were first considered (because of this property) by Reeve [16]in case d = 3 and have since been termed Reeve simplices . There are results that relate themaximal volume of an elementary polytope in R d to its surface area [5] or its inradius [2].Given a full-dimensional lattice polytope Π ⊆ R d and a set T of full-dimensional latticesimplices S , ..., S m ⊆ Π with m [ i =1 S i = Π and S i ∩ S j is a face of both S i and S j , ∀ i, j, then the set T = { S , ..., S m } is called a triangulation of Π. The triangulation T is called elementary or primitive if all its elements S i are, respectively, elementary or primitive simplices.Here is our main result on the equality (4): Proposition 2.3
If a full-dimensional lattice polytope Π ⊆ R d possesses a primitive triangula-tion then equality (4) holds for all n ∈ N . Proof.
Let n ∈ N and x ∈ n ∗ (Π ∩ Z d ). Then x = p + ... + p n for some p , ..., p n ∈ Π ∩ Z d , sothat x ∈ Z d and x = np with p = ( p + ... + p n ) /n . But p ∈ Π by convexity of Π.Now let x ∈ ( n Π) ∩ Z d , so x = np is an integer point with p ∈ Π. Let T = { S , ..., S m } bea primitive triangulation of Π and let w , ..., w d be the vertices of a simplex S i that contains p . Now there is a unique way to write p as a convex combination of w , ..., w d . So there are α , ..., α d ∈ [0 ,
1] so that p = α w + ... + α d w d and α + ... + α d = 1. Together with x = np thisimplies | | | w w · · · w d | | | · · · nα nα ... nα d = n | p | = | x | n . (5)If we refer to the matrix in (5) as M then, after subtracting the first column from all the othersand then expanding by the last row,det M = det | | | w w − w · · · w d − w | | | · · · = ( − d +1 det | | w − w · · · w d − w | | ∈ {± }
4y Lemma 2.1 since w − w , · · · , w d − w span the primitive simplex S i − w . So M − existsand is an integer matrix. By (5), it follows that β := nα , · · · , β d := nα d are integers since M − and x have integer entries. Summarizing, we get that β w + β w ... + β d w d = x, (6)where β , ..., β d ∈ { , ..., n } and β + ... + β d = n , so that (6) is the desired decomposition of x into a sum of n elements from Π ∩ Z d .It is not clear to us whether the existence of a primitive triangulation is necessary for equality(4) to hold for all n ∈ N . (Is it possible that every p ∈ Π is contained in a primitive simplex S ( p ) ⊂ Π without the existence of a “global” primitive triangulation of Π?)It is not hard to see that every lattice polytope Π possesses an elementary triangulation.(Existence of a triangulation can be shown by induction over the number of vertices of Π, andevery non-elementary simplex S i can be further triangulated with respect to its integer non-vertex points.) Existence of a primitive triangulation however is a different question – at leastin dimensions d ≥ Corollary 2.4 If Π is a full-dimensional lattice polytope in R d with d ∈ { , } then equality (4) holds for all n ∈ N . Proof.
Every lattice polytope has an elementary triangulation. In dimensions d ∈ { , } , byLemma 2.1 b), an elementary triangulation is always primitive. Now apply Proposition 2.3.In dimension d ≥
3, it is generally a difficult question whether or not a given lattice polytopeΠ has a primitive triangulation (see e.g. [4, 10]). The simplices Σ d,m in Example 2.2 a) with m ∈ { , ..., d − } and Σ ′ d,m in 2.2 b) with m ∈ { , , ... } are examples of lattice polytopes thathave no primitive triangulation. They are also examples, where (4) is not valid for general n ∈ N .For example, 2Σ , contains e , which cannot be written as the sum of two integer points fromΣ , . It is even possible to give examples Π where, for a given k ∈ N , (4) starts to fail at n > k while holding true for n = 1 , , ..., k . As an example, take Π = Σ k +1 , .The more specific question posed in the introduction is whether the fact that Ω := Π ∩ Z d (i) contains the origin, (ii) is symmetric (i.e. − x ∈ Ω if x ∈ Ω), and (iii) generates Z d , i.e. [ n ∈ N n ∗ Ω = Z d , (7)guarantees equality (4) for all n ∈ N . But also that has to be answered in the negative, asthe example Π = conv(Σ , ∪ − Σ , ) = conv {± e , ± e , ± ( − , − , ⊤ } ⊆ R shows. Indeed,( − , − , ⊤ is in 2Π but not in 2 ∗ Ω with Ω = Π ∩ Z = {± e , ± e , ± e , ± ( − , − , ⊤ , } .After all these examples, here are some results on the positive side:The symmetric hypercube [ − , d is the union of 2 d shifted copies of [0 , d , each of which hasa primitive triangulation. The cross-polytope conv {± e , ..., ± e d } triangulates into 2 d unit sim-plices around the origin. Much more involved, there is the following result by Kempf, Knudsen,Mumford and Saint-Donat [12]: Lemma 2.5
For every lattice polytope Π ⊆ R d , there is an integer k ∈ N such that k Π possessesa primitive triangulation. k ∈ N such that k Π satisfies (4) in placeof Π for all n ∈ N . Remark 2.6
The two conditions (7) with Ω := Π ∩ Z d , which says that Ω generates all of Z d ,and 0 ∈ int(Π), which implies that n ∗ Ω ⊆ ( n + 1) ∗ Ω, are connected with each other and withthe validity of (4) for all n ∈ N . Firstly, if (4) holds for all n ∈ N and 0 ∈ int(Π) then [ n ∈ N n ∗ (Π ∩ Z d ) = [ n ∈ N ( n Π) ∩ Z d = [ n ∈ N n Π ! ∩ Z d = Z d so that (7) holds. On the other hand, from (7) it follows, by the trivial inclusion “ ⊇ ” in (4),that Z d = [ n ∈ N n ∗ (Π ∩ Z d ) ⊆ [ n ∈ N ( n Π) ∩ Z d = [ n ∈ N n Π ! ∩ Z d ⊆ Z d , whence, by the convexity of Π, ∪ n Π = R d and hence 0 ∈ int(Π). Γ Let Γ be a finitely generated discrete group with identity element e . We are going to introducesome notions of topological type. Note that the standard topology on Γ is the discrete one; soevery subset of Γ is open with respect to this topology.Let Ω be a finite subset of Γ which contains the identity element e and which generates Γas a semi-group, i.e., if we set Ω := { e } and if we let Ω n denote the set of all words of lengthat most n with letters in Ω for n ≥
1, then ∪ n ≥ Ω n = Γ. Note also that the sequence (Ω n ) isincreasing; so the operators P Ω n can play the role of the finite section projections P Y n from theintroduction, and in fact we will obtain some of the subsequent results exactly for this sequence.With respect to Ω, we define the following “algebro-topological” notions. Let A ⊆ Γ. A point a ∈ A is called an Ω -inner point of A if Ω a := { ωa : ω ∈ Ω } ⊆ A . The set int Ω A of all Ω-innerpoints of A is called the Ω -interior of A , and the set ∂ Ω A := A \ int Ω A is the Ω -boundary of A . Note that we consider the Ω-boundary of a set always as a part of that set. (In this point,the present definition of a boundary differs from other definitions in the literature; see [1] forinstance.) One easily checks thatΩ n − ⊆ int Ω Ω n ⊆ Ω n and ∂ Ω Ω n ⊆ Ω n \ Ω n − (8)for each n ≥ ∂ Ω Ω n : • The sequence ( P ∂ Ω Ω n ) n ≥ belongs to the C ∗ -algebra S ( Sh (Γ)) which is generated by allfinite sections sequences ( P Ω n AP Ω n ) n ≥ where A runs through the operators on l (Γ) ofleft shift by elements in Γ (i.e., they are given by the left-regular representation of Γ on l (Γ)), and it generates the quasicommutator ideal of that algebra. • There is a criterion for the stability of sequences in S ( Sh (Γ)) which can be formulated bymeans of limit operators, and it turns out that it is sufficient to consider limit operatorswith respect to sequences taking their values in the boundaries ∂ Ω Ω n .6or details, see [19]. In many instances one observes that the “algebraic” boundary ∂ Ω Ω n coincides with the “numerical” boundary Ω n \ Ω n − ; in fact, one inclusion holds in general asmentioned in (8). We will see now that the reverse inclusion can be guaranteed if Γ = Z d andif Ω arises from a lattice polytope Π such that (4) holds for all n ∈ N . Proposition 3.1
Let Π be a lattice polytope in Z d which satisfies (4) and set Ω := Π ∩ Z d .Then ∂ Ω ( n ∗ Ω) = ( n ∗ Ω) \ (( n − ∗ Ω) (9) holds for all positive integers n . Proof.
As mentioned above, it is sufficient to show that( n ∗ Ω) \ (( n − ∗ Ω) ⊆ ∂ Ω ( n ∗ Ω) . We start with working on the continuous level and check first the implicationIf x ∈ n Π \ ( n − , then Π + x n Π . (10)Indeed, write x as tω with ω ∈ ∂ Π (= the usual topological boundary of Π) and n − < t ≤ n .Then x + ω = ( t + 1) ω with t + 1 > n , whence x + ω n Π.In the next step we show that ω can be chosen such that x + ω becomes a grid point for x a grid point. Indeed, let x ∈ ( n Π \ ( n − ∩ Z d . Consider the points x + ω i where the ω i , i = 1 , . . . , k , run through the (integer) vertices of Π. If we would have x + ω i ∈ n Π for each i ,then we would have x + Π = conv { x + ω i : i = 1 , . . . , k } ⊆ n Πby convexity of Π, which contradicts (10). Hence, for each x ∈ ( n Π \ ( n − ∩ Z d , there is avertex ω i of Π such that x + ω i ∈ (( n + 1)Π \ n Π) ∩ Z d . Employing the assumption (4) we conclude that, for each x ∈ n ∗ Ω \ ( n − ∗ Ω there is a ω i ∈ Ωsuch that x + ω i ∈ ( n + 1) ∗ Ω \ n ∗ Ω. Hence, x is in the Ω-boundary of n ∗ Ω.We proceed with an example which shows that the generalized version of (9), ∂ Ω Ω n = Ω n \ Ω n − , (11)does not hold for general subsets Ω of a finitely generated discrete group Γ and n ∈ N . Considerthe matrices ω := (cid:18) (cid:19) , ω := (cid:18) (cid:19) , ω := (cid:18) (cid:19) and ω := ω ω = (cid:18) (cid:19) , ω := ω − = (cid:18) −
10 1 (cid:19) , ω := ω ω = (cid:18) − (cid:19) . Then Ω := { ω i : i = 0 , . . . , } generates the group GL (2 , Z ) as a semi-group (clearly, Ω is notminimal as a generating system: ω , ω , ω , ω already generate this group). One easily checksthat ω ω = ω , ω ω = ω , ω ω = ω , ω ω = ω , ω ω = ω and ω ω = ω , whenceΩ ω ⊆ Ω. Thus, ω ∈ Ω \ Ω , but ω ∂ Ω Ω . So, (11) is violated already for n = 1.7et us conclude with a curious consequence of the coincidence (9) of the boundaries. Wementioned above that the sequence ( P ∂ Ω Ω n ) n ≥ always belongs to the C ∗ -algebra S ( Sh (Γ)) gen-erated by the finite sections sequences ( P Ω n AP Ω n ) n ≥ where A is constituted by operators of leftshift by elements of Γ. Under the conditions of Proposition 3.1, we conclude that the sequence( P Ω n − P Ω n − ) belongs to S ( Sh ( Z d )). In particular, the sequence ( P Ω n − ) = ( P Ω n ) − ( P Ω n − P Ω n − )belongs to S ( Sh ( Z d )). Consequently, with each sequence ( P Ω n AP Ω n ) n ≥ , the sequence( P Ω n − ) ( P Ω n AP Ω n ) ( P Ω n − ) = ( P Ω n − AP Ω n − )(with the operators P Ω n − AP Ω n − considered as acting on the range of P Ω n ) also belongs to S ( Sh ( Z d )). In particular, the algebra S ( Sh ( Z d )) contains a shifted copy (hence, infinitely manyshifted copies) of itself. The same fact clearly holds for every algebra which is generated by finitesections sequences ( Q n AQ n ) and contains the sequence ( Q n − Q n − ). A less trivial examplewhere this happens is the algebra of the finite sections method for operators in (a concreterepresentation of) the Cuntz algebra O N with N ≥ References [1]
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Authors:
Marko Lindner [email protected]
TU ChemnitzFakult¨at MathematikD-09107 ChemnitzGERMANYSteffen Roch [email protected]@mathematik.tu-darmstadt.de