On moments of a polytope

We show that the multivariate generating function of appropriately normalized moments of a measure with homogeneous polynomial density supported on a compact polytope P in R^d is a rational function. Its denominator is the product of linear forms dual to the vertices of P raised to the power equal to the degree of the density function. Using this, we solve the inverse moment problem for the set of, not necessarily convex, polytopes having a given set S of vertices. Under a weak non-degeneracy assumption we also show that the uniform measure supported on any such polytope is a linear combination of uniform measures supported on simplices with vertices in S.


Introduction
The initial motivation for the present paper came from proposed in [16] efficient algorithm recovering an arbitrary convex polytope from axial moments of a polynomial measure supported on it. This algorithm is based on the formulas for the axial moments of polytopes found over 20 years ago independently by M. Brion, J. Lawrence, A. Khovanskii-A. Pukhlikov, and A. Barvinok [6,11,18,23], see [7,8] for accessible explanation. In [16] the authors made an essential, although implicit, use of a univariate rational generating function for appropriately normalized axial moments. Here a multivariate, and explicit, analog of the latter function is developed. It turns out it provides a very convenient encoding of non-convex polytopes, which is of independent interest. E.g. it leads to a natural definition of vertices of such non-convex polytopes, which have similar properties to vertices of convex polytopes. It also allows to find the exact solutions of a class of inverse moment problems on non-convex polytopes.
Notation. In what follows we shall always assume that R d is endowed with a fixed coordinate system (x 1 , ..., x d ), orthonormal with respect to the standard scalar product ·, · . Let µ be a finite complex-valued Borel measure in R d . (For standard measure-theoretic notions we follow [25].) Given a multiindex I = (i 1 , . . . , i d ), let x I be the shorthand of the monomial x i1 1 . . . x i d d and |I| the shorthand for i 1 +· · ·+i d . For any multiindex I, define the moment m I (µ) of µ as Define the normalized moment generating function F µ (u) = F µ (u 1 , . . . , u d ) of µ by (|I| + d)! i 1 ! · · · i d ! m I (µ)u I , where u I = u i1 1 . . . u i d d .
( 1.2) Note that F µ (u) admits the integral representation which is a special case of a Fantappiè transformation. For details on the latter, see e.g. [4,Chapter 3]. A proof of (1.3) will be given at the end of Section 2; see also Remark 10. Given any complex-valued finite measure µ and any degree δ homogeneous dvariate polynomial ρ, it is convenient to define the (re)normalized moment generating function F ρ µ (u) for the measure ρµ, where by definition, R d f d(ρµ) = R d f ρdµ, in such a way that it can be obtained from F µ (u) by application of the differential operator ρ ∂ ∂u . Namely, set (|I| + d + δ)! i 1 ! · · · i d ! m I (ρµ)u I . (1.4) Note that F ρ µ (u) = F ρµ (u) for non-constant ρ. However, they are also connected, by an explicit differential operator as follows.
Theorem 1. For any complex-valued finite measure µ and any homogeneous polynomial ρ of degree δ, (1.7) Here and in what follows • denotes the application of a differential operator to a function. The proof of the latter result is basically an exercise in manipulating formal power series, and we do not claim its novelty. For the sake of completeness, we include a proof in Section 2.
Results on convex polytopes. A finite set S ⊂ R d is called spanning if it is not contained in any (affine) hyperplane in R d . (Obviously, card(S) ≥ d + 1.) As usual, by a (compact, convex) polytope P ⊂ R d we mean the convex hull of a finite spanning set in R d . The set of vertices of a convex polytope P is the inclusionminimal finite set with convex hull P. A d-simplex in R d is the convex hull of a spanning (d + 1)-tuple of points. By an open polytope (resp. simplex) we mean the set of interior points of a compact polytope (resp. simplex).
Given a convex polytope P let V = (v 1 , ..., v N ) denote the set of its vertices. Assume that P is simple, i.e. each v ∈ V has exactly d incident edges vv e1 , . . . , vv e d . Set w k (v) := v e k − v, for 1 ≤ k ≤ d. The non-negative real span K v of w 1 (v),. . . , w d (v) is called the tangent cone of P at v. For each K v define | det K v | = | det(w 1 (v), . . . , w d (v))| to be the volume of the parallelepiped formed by w 1 (v), . . . , w d (v).
Given a bounded domain Ω ⊂ R d , we call the measure where χ Ω is the characteristic function of Ω, the standard measure of Ω. For a simple convex polytope P we have the following explicit representation of F µP (u).

Theorem 2.
For an arbitrary simple convex polytope P, (1.9) Remark 1. Instead of the explicit choice of w k (v) for v ∈ V made above, we can take any fixed set of non-zero vectors w 1 (v), . . . , w d (v), spanning the tangent cone of v in P. This does not affect the validity of (1.8) and (1.9).

Theorem 2 implies
Corollary 3. Let ∆ = conv(V) ⊂ R d be an arbitrary d-simplex. Then (1.10) Remark 2. As we discovered after we proved the above results, statements similar to Corollary 3 in the complex setting can be found in [4,Section 3.5] and in particular [4, Corollary 3.5.6]. A variation of (1.10) also appears in [5], in the context of designing an efficient procedure for integration of polynomials over simplices.
Notice that an arbitrary convex polytope P admits a triangulation which only uses the existing vertices of P, see e.g. [8,Theorem 3.1]. Applying Corollary 3 and Theorem 1 to the sum of measures corresponding to such a triangulation we get the following.
Corollary 4. The normalized moment generating function F ρ P (u) of any convex polytope P with respect to any homogeneous polynomial density function ρ of degree δ is a rational function with denominator dividing Example 1. Let ∆ be a triangle in R 2 with vertices v 1 = (1, 1), v 2 = (2, 5) and v 3 = (3, 2). Its normalized moment generating function equals Its Taylor expansion about the origin up to the terms of degree 7 is given by Results on non-convex polytopes. Our second group of results addresses the problem of distinguishing different polytopes with the same underlying set of vertices from information on their moments. The problem of restoring the vertices of a polygon or a polytope with a constant mass density from information on its moments was addressed earlier in e.g. [12, 15-17, 20, 27]. However, the latter do not provide the recovery of the vertices in the generality required in the present paper. Below we concentrate on the case of constant density and known vertices, and plan to return to the general inverse problem for polytopes with unknown polynomial density and unknown location of their vertices in the future. First we need to define what we mean by a polytope. It turned out that there is no general consensus about this notion. Instead there exist several competing definitions having their own advantages in different situations. We shall study the following class of polytopal objects. Definition 1. A subset P ⊂ R d coinciding with a finite union of arbitrary convex d-dimensional polytopes is called a generalized polytope.
Definition 2. The number of components of a generalized polytope P is the number of connected components of the set P o ⊂ P of interior points of P. The closure of each connected component of P o is called a component of P. A generalized polytope with one component is called indecomposable.
Remark 3. We say that a simplicial complex in R d is pure if all its maximal simplices have dimension d. Clearly any generalized polytope in R d can be represented as the topological space of an appropriate pure simplicial complex.
Remark 4. Often one considers a more restricted class of objects, namely polytopes. A polytope P ⊂ R d is a generalized polytope homeomorphic to a d-dimensional manifold with boundary.
We need to introduce the notion of a vertex of a generalized polytope. A wealth of material on dissections of polytopes can be found in [21], see also [13].
Definition 4. Given a generalized polytope P ⊂ R d we call a point v a vertex of P, if v is a vertex of (the closure of) some open simplex in every dissection of P.
Definition 5. Given a point p ∈ P of a generalized polytope P we denote by the tangent cone T p (P) of P at p the set obtained as follows. For a sufficiently small ǫ > 0 set P p (ǫ) = P ∩ B p (ǫ) where B p (ǫ) is the ǫ-ball centered at p. Define T p (P) as the set obtained by taking a ray though p and every point of P p (ǫ). In other words, T p (P) is the cone with the apex at p and the base B p (ǫ). (Obviously, T p (P) is independent of ǫ for a sufficiently small ǫ > 0.) does not admit a decomposition in the disjoint union of convex polygonal subcones, such that each subcone in the decomposition has a translation-invariant direction. In particular, if the tangent cone to P at v has a connected component with no translation-invariant direction then v is a vertex.
We denote by conv(S) the convex hull of an arbitrary set S ⊂ R d . The above lemma implies that any vertex of conv(P) is a vertex of P.
The following result extends Corollary 4 to the case of generalized polytopes.
Proposition 6. For any generalized polytope P with the set of vertices V(P), the denominator of its normalized moment generating function F ρ P (u) with respect to a homogeneous polynomial density function ρ of degree δ divides Remark 5. There exist generalized polytopes which do not admit dissections with only existing vertices. The simplest example of this kind is the Schönhardt polyhedron, see Figure 1 and [26]. Absence of a dissection T which uses only its 6 vertices can be established by observing that none of the edges AC, A ′ B, and B ′ C ′ can appear in a simplex of T , yet any simplex on these 6 vertices must contain one of them. Therefore, Proposition 6 is not an immediate consequence of Corollary 3. Remark 6. For "generic" generalized polytopes P the denominator Ω(u) of F P (u) equals Φ P (u), but for certain special polytopes the denominator Ω(u) may be its proper divisor, as can be seen from the following example. Let A = {0, a 1 , a 2 , a 3 } ⊂ R 3 be a spanning set, and v ∈ R 3 . Let P ± := conv(v ± A) and P := P + ∪ P − . Then 1 − u, v does not appear in Ω(u), as where K = 0 is a real constant.
Now we introduce several finite-dimensional linear spaces related to a given finite spanning set S ⊂ R d . Let P(S) be the set of all generalized polytopes P whose sets V(P) of vertices are contained in S. For P ∈ P(S) we denote by µ P its standard measure. (Obviously, µ P is supported on P ⊆ conv(S).) Denote by M(S) the linear space of all signed measures, i.e. the linear span of all standard measures µ P for P ∈ P(S). Let M ∆ (S) ⊆ M(S) be its subspace spanned by µ ∆ , for ∆ ∈ P(S) a d-dimensional simplex. (The space M ∆ (S) has earlier appeared in [1], [2], [3] in a somewhat different context.) We shall refer to elements of M(S) as to polytopal measures with the vertex set S. The next conjecture is central to our study.

Conjecture 7.
For an arbitrary spanning set S and any generalized polytope P with set of vertices contained in S, its standard measure µ P belongs to M ∆ (S). In other words, M(S) = M ∆ (S).
By Remark 5, the above conjecture is non-trivial. While we do not have a proof of Conjecture 7 in its full generality, we have succeeded in proving it for a rather large class of spanning sets. Roughly speaking, the latter should be close to "generic". Specifically, given a finite spanning set S ⊂ R d , we say that S is weakly non-degenerate if any (d + 2)-tuple of points from S is spanning. If S satisfies the stronger condition that each (d + 1)-subset of S is spanning then we call the latter S strongly non-degenerate.
Theorem 8. Conjecture 7 holds for any weakly non-degenerate finite set S.
Remark 7. Theorem 8 would imply Conjecture 7 if one could prove that the standard measure of an arbitrary generalized polytope P can be obtained as the limit of the standard measures of a 1-parameter family of generalized polytopes P(t) with P(0) = P such that for t = 0 the vertices of P(t) are weakly non-degenerate, and each vertex of P(t) tending to a vertex of P as t → 0. Unfortunately we are unable to prove the existence of such deformations in general.
The key idea in the proof of Theorem 8 is to study the corresponding spaces of Fantappiè transformations of signed measures in M(S). In particular, we are able to compute the corresponding dimensions 1 . In more detail, let F(S) (resp. F ∆ (S)) be the linear space of Fantappiè transformations of signed measures in M(S) (resp. M ∆ (S)). In other words, F(S) (resp. F ∆ (S)) is the space of normalized moment generating functions of signed measures in M(S) (resp. M ∆ (S)).
Since each compactly supported measure is uniquely determined by its complete set of moments, the map  Corollary 10 implies that for strongly non-degenerate S the dimension of all these linear spaces equals N −1 d . Note that Corollary 10 settles Theorem 8 for the strongly non-degenerate S.
Our final goal is to explicitly solve the following inverse moment problem. Problem 1. Given a strongly non-degenerate spanning set S ⊂ R d , |S| = N , find the unique polytopal measure in M(S) with a given set of all moments up to order N − d − 1.
We start with the following simple observation.   Theorem 12. For an arbitrary strongly non-degenerate spanning set S ⊂ R d , |S| = N , the matrix M at S is invertible. Moreover, for a rational function R(u) = P (u)/Φ S (u), where P (u) is an arbitrary polynomial of degree N − d − 1, there exists a unique measure µ R ∈ M(S) with Fantappiè transform R(u). Namely, Remark 8. A detailed explanation of the meaning of (1.13) can be found in the proof of Theorem 12, see also Example 2 below. An explicit formula for the matrix M at −1 S is given in Lemma 18.
Recall that a spanning set S is weakly non-degenerate if any (d + 2)-tuple of its points is spanning. With minor changes, the above solution of the inverse moment problem can be adapted to this more general case. In order not to overload the introduction we refer the readers interested in this situation to Section 4. The case of an arbitrary spanning set S, however, remains unsolved and offers several interesting challenges in matroid theory. We hope to return to it in the future.
It will be convenient to work with scaled volumes of simplices, which we call weights.
Definition 6. Given a signed measure µ in R d and a d-dimensional simplex ∆ ⊂ R d we define the weight w ∆ of ∆ by the formula: (1.14) In other words, the density d ∆ of the measure in question which should be placed at ∆ equals We finish the introduction by explicitly solving the above inverse problem for a concrete 5-tuple of points in R 2 .
To illustrate all steps of solution of our inverse moment problem assume that we are looking for a polygonal measure with the vertex set S and (ad hoc chosen) moments m 00 = 1, m 10 = 2, m 01 = 3, m 20 = 4, m 11 = 5, m 02 = 6. Then its normalized moment generating function F µ (u) satisfies the relation where P (u 1 , u 2 ) is a (non-homogeneous) polynomial of at most second degree. Thus, truncating the product of the left-hand side and l 1 l 2 l 3 l 4 l 5 up to the second degree, we obtain  Remark 9. Domains into which the convex hull conv(S) is cut by the hyperplanes spanned by S were introduced in [1] where they were called chambers. The incidence matrix of the simplices spanned by S and those chambers was studied in some detail in [2], [3]. This matrix allows to formalize the last step of construction of the above polygonal measure, where information on the densities of the simplices is transformed into information on the densities of the chambers. But, in general, already the number of chambers is a complicated invariant of the set S. It seems that the general problem of constructing the set of chambers and the corresponding incidence matrix in terms of a given S is quite non-trivial.
Acknowledgement. The second author is grateful to the Mathematics Department of Stockholm University for the hospitality in June 2011 when this project was initiated. The third author wants to acknowledge the hospitality of the School of Physical and Mathematical Sciences, Nanyang Technological University in April 2012 when this project was completed. We want to thank Sinai Robins for numerous discussions of the topic. We acknowledge extremely helpful answers and comments on our questions on mathoverflow.net, in particular ones by David Eppstein, Dirk Lorenz, Igor Pak, David Speyer, and Gjergji Zaimi. Finally, the third author wants to thank late Mikael Passare (who unfortunately left us so early) for discussions of the properties of Fantappiè transformation and for pointing out reference [4] in September 2011.

Proving results on convex polytopes
Following Brion-Lawrence-Khovanskii-Pukhlikov-Barvinok, see [6,8,16,18], we define for each vector z ∈ R d the j-th axial moment µ j (z) of a simple convex polytope P with respect to z as We will use the following important statement, cf. e.g. [8,Theorem 10.5].
,z , and z is an arbitrary vector for which the products d j=1 w j (v), z , v ∈ V, do not vanish. Moreover, the following identities hold: Proof of Theorem 2. To prove (1.8), consider the generating function where u = uz. On the other hand, using the multinomial coefficients |J| and (1.8) follows.
In view of relations (2.2) the right-hand side of (1.8) can be rewritten as (1.9).
j=0 v, u j and expanding (1.8) with respect to jth powers of v, u , we see that (2.2) implies that for j < d the sum of all terms v, u j vanishes.
The right-hand side of (1.8) becomes Computing the common denominator of the latter, we obtain .
It is convenient to introduce one more linear form ζ d+1 := 1, so that the last expression reads as To complete the proof we notice that Indeed, the first matrix has two identical rows and thus vanishing determinant, which we expand with respect to the last row. The last equality is the standard formula for the Vandermonde determinant. Thus we have
In [22] a similar idea was applied to the harmonic polygonal measures in the plane.
Proof of Theorem 1. Assume first that ρ(x 1 , ..., x d ) = x K = x k1 1 · · · x k d d is a monomial and consider One gets Observe that the normalizing coefficients of m I (x K µ) in the latter expression depend only on I and |K| but not on particular entries of K. Therefore for an arbitrary homogeneous ρ of degree δ one gets by additivity This shows (1.6). Repeated application of (2.4), for d + 1 ≤ ℓ ≤ d + δ, to the integral representation (1.3), respectively, to the representation (1.2), of F ρµ (u) implies (1.7), respectively, (1.5).

Inverse moment problem for strongly non-degenerate S
Proof of Lemma 5. We prove first that the tangent cone at any non-vertex allows a decomposition into convex polytopal cones each having a translation-invariant direction.
Let v be a point in P which is not a vertex. Then there is a dissection T of P such that v is not a vertex of any simplex of T . Let U be the set of simplices S u of T with closures containing v. Take the dissection of the tangent cone T v (P) into the tangent cones to simplices from U , T v (P) = ∪ u∈U T v (S u ). Clearly, every subcone T v (S u ) contains a translation-invariant direction (any direction parallel to the minimal face containing v).
Vice versa, to prove the converse implication, let us take a dissection of the tangent cone T v (P) into a disjoint union of convex polytopal cones Q 1 , . . . , Q k . By definition of the tangent cone and since P can be represented as a finite union of simplices we obtain that any sufficiently small neighborhood of v in the tangent cone T v (P) is a neighborhood of v in the entire P. Consider the parallelepiped Box ε centered at v that is ε ball centered at v, in the L 1 -norm. Note that each convex polytopal set Q i ∩ Box ε can be decomposed into a union of simplices that do not contain v as a vertex.
Further notice that the set P \ Box ε can be represented as a finite disjoint union of simplices, since Box ε is the intersection of a finite number of half-spaces and P is a disjoint union of simplices. Clearly, every simplex in this union should not have v as a vertex. Now combining the dissections of each Q i ∩ Box ε and P \ Box ε we obtain the required dissection of P.
Proof of Proposition 6. We begin by considering the case ρ ≡ 1. Let T be a dissection of P with vertices V(T ). Corollary 3 implies that F P (u) has a denominator dividing g T (u) = v∈V(T ) (1 − v, u ). Take v 1 ∈ V(T ) \ V(P). Then there exists another dissection T ′ such that v 1 ∈ V(T ′ ). Expressing F P (u) as ratios of polynomials, we have Here g T ′ is not divisible by 1 − v 1 , u , by the choice of T ′ . Thus f T is divisible by 1 − v 1 , u , and can be canceled out in the expression for F P (u). The case of arbitrary homogeneous ρ follows immediately by applying Theorem 1 to the already covered case ρ ≡ 1.
Proof of Proposition 9. First we show that for an arbitrary finite spanning set S ⊂ R d the space M ∆ (S) has a basis of d-dimensional simplices containing a fixed vertex v ∈ S. In particular, the set of all d-dimensional simplices containing v spans M ∆ (S) but is not necessarily a basis. Consequently, their Fantappiè transformations spans F ∆ (S). The following result is formulated as Theorem 4.2 of [2] and in a different form in [1]. (We omit the proof of this statement here.) Given two points p and q and a set M in R d , we say that q is visible from p with respect to M if the straight interval pq is disjoint from M .
where µ σi,p is the standard measure of the d-dimensional simplex spanned by the vertices of σ i and the point p.
Remark 11. If σ i,p is a degenerate simplex, i.e. p lies in the hyperplane spanned by σ i , we simply exclude the corresponding term µ σi,p from the above formula.
To prove Proposition 9, we need to show that F ∆ (S) coincides with Rat(S) if and only if S is strongly non-degenerate. Indeed, F ∆ (S) ⊆ Rat(S) for an arbitrary spanning S, by Proposition 6. The Fantappiè transform Proof. We recall that F ∆ (S) comprises all linear combinations of the rational func- .
For the ntuple L = {l 1 , l 2 , ...., l n } of linear (d+1)-variate forms, let V L be the linear span of all possible products of the form l i1 l i2 . . . l i n−d , 1 ≤ i 1 < i 2 < ... < i n−d ≤ n. Observe that V L is the space of all numerators that one can obtain in F ∆ (S). We need to show that V L contains HP oly(n−d, d+1), the space of all d+1-variate homogeneous polynomials of degree n − d. Recall that any d + 1-tuple of linear forms l i1 , . . . , l i d+1 is linearly independent due to the strong degeneracy assumption. Thus we can express each single variable u 0 , . . . , u d as a linear combination of these forms. Since V L contains all products l i1 l i2 . . . l i n−d−1 l j , where j ∈ {1, . . . , n} \ {i 1 , . . . , i n−d−1 }, we conclude that V L contains all homogeneous polynomials of the form From that we deduce that V L contains all homogeneous polynomials of the form l i1 l i2 . . . l i n−d−2 u k u j , where j, k ∈ [n] \ {i 1 , . . . , i n−d−1 }. Continuing along the same lines we derive by induction that V L contains HP oly(n − d, d + 1).
For an arbitrary spanning S the cardinality of B N is at most N −1 d = dim Rat(S). Furthermore, if S is not strongly non-degenerate the cardinality of B N is strictly smaller than N −1 d , as there will be linear dependencies among the standard measures on the simplices in B N . Therefore dim F ∆ (S) < dim Rat(S).
We define the square matrix M at L of size n d with entries being coefficients of the above products of linear forms w.r.t. the standard monomial basis in HP ol(n − d, d + 1).
Lemma 17. The determinant of M at L is proportional to the product of the determinants of all (d + 1)-tuples (l i1 , l i2 , . . . , l i d+1 ), i 1 < i 2 < . . . < i d+1 . (By the determinant of a (d + 1)-tuple of vectors in R d+1 with a fixed basis we mean the determinant of the matrix formed by the coordinates of these vectors in a chosen basis.) Proof. Indeed, det(M at L ) is a form of degree (d + 1) n d in the coefficients of the linear forms l 1 , . . . , l n . Thus the product i1,...,i d+1 det(l i1 , l i2 , . . . , l i d+1 ) has the same degree as det(M at L ). Therefore it suffices to show that det(M at L ) vanishes as soon as some of det(l i1 , l i2 , . . . , l i d+1 ) vanishes. (Observe that all polynomials det(M at L ) are coprime.) Without loss of generality assume that l 1 is a linear combination of l 2 , . . . , l d+1 . But then the column of M at L corresponding to the (n−d)-tuple (1, d+2, d+3, . . . , n) will be a linear combination of those corresponding to (2, d + 2, d + 3, . . . , n), . . . (d + 1, d + 2, d + 3, . . . , n). To solve the latter inverse problem using Corollary 3, we need to find an appropriate set of weights w = {w i1,...,i d }, .
Clearing the denominators, we get the equation We also need the following (d + 1) given by: i.e. this holds for all terms in the product on the right-hand side of (3.5). Hence, if e = e ′ then e T ·(M at S · e ′ ) = 0, as among On the other hand, if e and e ′ coincide, then the right-hand side of (3.5) is equal . Dividing by the common denominator of the fractions in e, we obtain e T · (M at S · e) = 1.

Inverse moment problem for weakly non-degenerate S
Given an arbitrary spanning set S = {v 1 , v 2 , . . . , v N }, consider the linear space The next statement explains why we can extend our solution of the inverse moment problem from the case of strongly non-degenerate S to the case of weakly non-degenerate S. Proof. We have N (non-homogeneous) linear forms l 1 . . . , l N in variables u = (u 1 , . . . , u d ) and the linear space V L spanned by all possible products of (N − d − 1)-tuples of distinct forms. We need to investigate whether V L coincides with P ol (N − d − 1, d). Homogenizing, we consider the same question for the linear homogeneous forms and the homogeneous polynomials of degree N − d− 1 in variables (u 0 , u 1 , . . . , u d ).
First assume that there are d+2 linear forms l 1 , . . . , l d+2 which are not spanning. Then one can find a non-zero vector z 0 ∈ R d+1 , such that l 1 (z) = · · · = l d+2 (z) = 0. Note that each product of N − d − 1 different forms chosen from l 1 , . . . , l N contains at least one form among {l 1 , . . . , l d+2 }. Therefore any linear combination of products of N − d − 1 forms vanishes at z 0 . Thus V L cannot coincide with HP ol (N − d − 1, d + 1).
Conversely, assume that every (d + 2)-tuple of distinct forms among l 1 , . . . , l N is spanning. First, we notice that HP ol(N − d − 1, d + 1) can be spanned by the all possible products of N − d − 1 linear forms (not necessarily pairwise distinct). Indeed, since first d + 2 forms span the dual space of R d+1 , we can express each variable x i as a linear combination of these forms. Therefore every monomial of degree N − d − 1 can be expressed as a linear combination of products of N − d − 1 forms. Now we show that each product of N − d − 1, not necessarily distinct, forms can be expressed as a linear combination of the products of distinct ones. Assume the contrary and consider monomials l i1 1 . . . l iN N of degree N − d − 1 which cannot be expressed as a linear combination of products with all distinct forms. Among those monomials we take a monomial m = l k1 1 . . . l kN N having the maximal number of distinct forms in the product. Since m is not a product of all distinct forms, it should contain a form l i in some power k i ≥ 2. Given that k i ≥ 2 and the degree of m is N − d − 1, one can find d + 2 distinct forms l i1 , . . . , l i d+2 that do not appear in m. Since any d + 2 of our forms span the dual space of R d+1 , we can express l i as a linear combination of l i1 , . . . , l i d+2 . Now rewrite m as α 1 · l i1 + · · · + α d+2 · l i d+2 l k1 1 . . . l ki−1 i . . . l kN N , where α 1 · l i1 + · · · + α d+2 · l i d+2 = l i . Thus we get an expression of m as a linear combination of monomials α j · l ij l k1 1 . . . l ki−1 i . . . l kN N , where each such monomial has more distinct forms than m. Each of such monomials can be expressed as a linear combination of products of all distinct forms, since m was chosen as a monomial with the maximal possible number of distinct forms, which cannot be expressed in such a way. This is a contradiction. Therefore m can also be expressed as a linear combination of products of all distinct forms.
Below we consider the inverse problem for a weakly non-degenerate S, using notation from (3.1) and (3.3). Here we no longer have a natural basis of all simplices sharing a common vertex v N . Because of that we need to consider all N points and include one more linear form l N into the corresponding matrix L. Slightly abusing our notation, we denote by L the same matrix as before, although it contains one more (last) column corresponding to v N . Similarly to notation Proof. Fix the set S as above. In what follows we treat both sides of (4.1) as complex-valued polynomials in N ·(d+1) variables, these variables being the entries of matrix L. We first show that every determinant L(J) divides det M at S (S) . Indeed, let J = {j 1 , . . . , j d+1 } be a set of (d + 1) forms which has a nonempty intersection with any (N − d − 1)-tuple of forms in S. Let z = (z 1 , . . . , z N ·(d+1) ) be a zero of the polynomial L(J), which means that forms l j1 , . . . , l j d+1 comprised of the corresponding coordinates of z are linearly dependent. Therefore, there is a nonzero vector u 0 ∈ R d+1 , such that l j1 (u 0 ) = · · · = l j d+1 (u 0 ) = 0. Consider the row vector (u I 0 ) consisting of N −1 d homogeneous monomials of degree N − d − 1 evaluated at u 0 . We notice that (u I 0 ) is in the kernel of M at S (S), as the product of (u I 0 ) with each column vector T ∈ S of M at S (S) is equal to j∈T l j (u 0 ); and every set T ∈ S contains at least one of the forms l j1 , . . . , l j d+1 in such a product.
Thus det M at S (S) also vanishes at such z.
We recall a well-known fact (cf. e.g. [10, Theorem 61.1]) that is an irreducible complex-valued polynomial in variables z j1 , z j2 . . . , z j d+1 +N d . Now if every zero of an irreducible polynomial p(z 1 , . . . , z N (d+1) ) annihilates another polynomial q(z 1 , . . . , z N (d+1) ) then p divides q. We conclude that L(J) divides det M at S (S) . Using the fact that each L(J) is an irreducible polynomial and all L(J)'s are pairwise distinct (i.e. have distinct sets of projective zeros) we conclude that the product of L(J)'s in the right-hand side of (4.1) divides det M at S (S) .
Finally, the product of L(J)'s has the degree We observe that for each T ∈ S the complementary set of d + 1 forms cannot be taken as a feasible J. We notice further that these complements are the only exceptions for the choice of J. Therefore as a feasible J we can pick any of N d+1 (d + 1)-tuples except those N −1 d complements of a T ∈ S. Therefore, (4.2) equals The can be interpreted as the normalized moment generating function of an appropriately scaled standard measure of the d-dimensional simplex spanned by these vertices.
If v i1 , v i2 , . . . , v i d+1 only span a hyperplane H in R d then (4.3) corresponds to a singular (w.r.t. to the Lebesgue measure on R d ) measure µ δ supported on δ = conv(v i1 , . . . , v i d+1 ). One way to define it as the weak limit of a sequence of (absolutely continuous with respect to the Lebesgue measure on R d ) measures-the appropriately scaled standard measures µ δt of family of d-dimensional simlices δ t which degenerate into δ when t = 0. There is no loss in generality in assuming K i1i2...i d+1 = 1, i.e. to deal with probability measures.
Proposition 21. Let W = {w 1 , . . . , w d , w d+1 } be a (d + 1)-tuple of points in R d such that W spans a hyperplane H ⊂ R d . Denote by l w1 = 1 − w 1 , u , . . . , l w d+1 = 1 − w d+1 , u the associated linear forms. There exists a unique measure µ W supported on δ = conv(W) with the normalized moment generating function F µW (u) given by Proof. Without loss of generality assume that W = {w 1 , . . . , w d , w d+1 } is ordered in such a way that {w 1 , . . . , w d } span H. Then, δ t is defined as δ t = conv(δ, w i d+1 +tz), with z a unit normal to H, and µ δt as the uniform density probability measure supported on δ t . Then lim f dµ δt = f dµ δ for any bounded, continuous real function on R d , cf. e.g. [9]. Then, this measure has compact support, and thus is determined by its moments, cf. e.g. [24, Proposition 3.2].
Remark 12. One can prove that the integration of a smooth compactly supported function φ with respect to the limiting measure µ W is given by the integration of φ over δ with a continuous piecewise linear weight function uniquely determined by δ. Similar limits appear frequently in the theory of splines. Since we only need the existence of µ W we do not pursue this topic here.
Our solution of the inverse moment problem for the linear space M(S) closely follows the pattern presented in Example 2. In other words, given a weakly nondegenerate S and the set of moments up to order N − d − 1 we (i) produce the rational function R(u) ∈ Rat(S) with Taylor coefficients coinciding with the normalized moments; (ii) represent R(u) in the form (4.3); (iii) for each term as in (4.4) determine the underlying measure supported on the (probably degenerate) convex hull of the vertices v i1 , v i2 , . . . , v i d+1 .
We can now prove our central result claiming that M ∆ (S) = M(S) for a weakly non-degenerate S.
Proof of Theorem 8. Theorem 8 is already settled in Corollary 10 for the case of strongly non-degenerate S. It remains to consider the case of weakly non-degenerate S. The denominator of the moment generating function F P (u) for an arbitrary generalized polytope P with the vertex set S is of the form Π N i=1 l i by Proposition 6, and its numerator belongs to P ol(N − d − 1, d). As S is weakly non-degenerate, F P (u) can be written as a linear combination of the fractions as in (4.3), where (i 1 , i 2 , . . . i d+1 ) runs over the set of (d + 1)-tuples of indices. If a (d + 1)-tuple l i1 , l i2 , . . . l i d+1 is spanning then is the moment generating function of the measure supported on the simplex ∆, determined by its denominator, with the uniform density K/d!Vol(∆). By Proposition 21, if a (d + 1)-tuple l i1 , l i2 , . . . l i d+1 is not spanning then is the moment generating function of a singular measure supported on a degenerate simplex. As P is a generalized polytope, its standard measure has no singular components. Therefore, no degenerate simplices can appear in its decomposition.
Remark 13. The latter proof demonstrates that if one starts from the set of moments of the standard measure µ of a polytope with the vertex set S then we never obtain degenerate simplices while solving the inverse moment problem. This is why M ∆ (S) = M(S). However, an explicit description of F ∆ (S) for a general weakly non-degenerate S is missing at present. For concrete Examples 3 and 4 we give these descriptions below.
Our final result computes dim M ∆ (S) and describes a procedure to construct a basis for M ∆ (S). are linearly independent, as each degenerate simplex defines a singular measure supported in a proper hyperplane, and these hyperplanes differ for different degenerate simplices. We are done with (i).
Let Σ 0 be a dependent d + 1-subset of S, and δ = conv(Σ 0 ) be as in (ii). Then δ spans a hyperplane H 0 . As each d-dimensional simplex on σ 0 := v i and d vertices from Σ 0 is uniquely defined by the latter, it suffices to analyze dependencies between the standard measures of d − 1-simplices with vertices in Σ 0 .
We can view Σ 0 as a weakly non-degenerate subset in R d−1 ∼ = H 0 . By (i), we . If Σ 0 is strongly non-degenerate as a subset of H 0 ∼ = R d−1 , i.e. ♯ deg (Σ 0 ) = 0, then dim M ∆ (Σ 0 ) = d, i.e. there is exactly one linear dependence between the standard measures of d − 1-simplices with vertices in Σ 0 , and we are done. Otherwise, Σ 0 = {σ 1 } ∪Σ 1 , with Σ 1 spanning a hyperplane H 1 in H 0 . Moreover, this can only happen if d ≥ 3. Now, we can repeat the whole argument with σ 1 in place of σ 0 , Σ 1 in place of Σ 0 , and H 1 in place of H 0 . Again, we either have Σ 1 strongly degenerate, and we are done, or we repeat this argument, etc., until we hit a strongly non-degenerate Σ k , which is bound to happen, as the dimension goes down each iteration. This completes the proof of (ii).
Then, (iii) stems from the fact that the vertices of d-dimensional simplex on v i distinct from v i span a hyperplane, and the only possibility for a degenerate simplex δ as in (ii) is to lie in this hyperplane.
Finally, to prove (iv), observe that the set B ′ i of the standard measures of d-dimensional simplices containing a given vertex v i always spans M ∆ (S), see Lemma 15. Now for each degenerate d-simplex δ we prune B ′ i by removing the standard measure of a simplex in the linear dependence corresponding to δ. In view of (ii) and (iii) this process is well-defined and unambiguous. In the end we obtain N −1 d − ♯ deg standard measures of d-dimensional simplices. In view of (i) they form a basis of M ∆ (S), as claimed.

Remark 14.
The above discussions show that the columns of M at S corresponding to degenerate simplices must necessarily be included in any non-vanishing maximal minor M at S (S).
We conclude our discussion of the weakly non-degenerate case with two examples.
Triangles ∆ 135 and ∆ 124 are degenerate which implies that if the original measure we are recovering is polygonal then w 135 = w 124 = 0. Therefore the linear space of numerators P (u 1 , u 2 ) for the space F ∆ (S) in this example is given by the relation a 01 = a 11 = a 02 .
Our last example is more degenerate than the previous one, although still weakly non-degenerate. In fact, in this example S is a multiset since v 1 = v 5 . It shows that our technique can be generalized to a certain class of multisets as well. Thus, given an arbitrary rational function R(u 1 , u 2 ) = P (u 1 , u 2 )/Φ S (u 1 , u 2 ) where P (u 1 , u 2 ) = a 00 + a 1,0 u 1 + a 0,1 u 2 + a 2,0 u 2 1 + a 11 u 1 u 2 + a 02 u 2 2 is any polynomial of degree at most 2 and Φ S (u 1 , u 2 ) = l 1 l 2 l 3 l 4 l 5 , we obtain Notice that triangles ∆ 125 , ∆ 135 , ∆ 145 , ∆ 234 are degenerate. If we know that the original measure we are recovering is polygonal then one should get w 125 = w 135 = w 145 = w 234 = 0. Therefore, the linear space of numerators P (u 1 , u 2 ) for the space F ∆ (S) in this example is given by the system of equations:          a 20 = 0 a 10 = a 11 a 02 = 0 4a 00 + 2a 01 + 3a 10 = 0.

Remarks and open problems
Remark 15. A weaker form of Corollary 4 (i.e. the rationality of F ρ P (u), but without the claim on the particular shape of the denominator) can be derived directly from (1.3) by using Stokes formula, along the lines of [6, Lemma 1].

Problem 2.
Find an appropriate version of Theorem 2, applicable to non-simple and/or non-convex polytopes.  Remark 17. If either P itself or its complement in conv(S) can be triangulated by simplices with vertices in S then µ P is an integer point in M Z (S).
Let us pose the following tantalizing question. Problem 3. Does there exist a generalized polytope P with the set of vertices S(P) and with non-integer coordinates in M Z (S(P))?
Problem 4. One can also define a rational convex cone Pos(S) ⊂ M Z (S) by taking non-negative linear combinations of all µ P , where P runs over the set of all generalized polytopes in P(S).
Conjecture 24. The rational cone Pos(S) is uniquely determined by the oriented matroid associated to S.
We conclude this section with the following question. One can easily show that a simplex from P(S) spans an extremal ray of Pos(S) if and only if it does not contain any points of S distinct from its vertices. Problem 5 is apparently closely related to the problem of classification of combinatorial types of point arrangements, see e.g. [14] and references therein.