# On moments of a polytope

Nick Gravin  School of Physical and Mathematical Sciences, Nanyang
Technological University, 21 Nanyang Link, 637371 Singapore
<EMAIL_ADDRESS>,  Dmitrii V. Pasechnik  School of Physical and
Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link,
637371 Singapore<EMAIL_ADDRESS>,  Boris Shapiro  Department of Mathematics,
Stockholm University, SE-106 91 Stockholm, Sweden<EMAIL_ADDRESS>and
Michael Shapiro  Department of Mathematics, Michigan State University, East
Lansing, MI 48824-1027, USA<EMAIL_ADDRESS>To the memory of Mikael
Passare

###### Abstract.

We show that the multivariate generating function of appropriately normalized
moments of a measure with homogeneous polynomial density supported on a
compact polytope $\mathcal{P}\subset\mathbb{R}^{d}$ is a rational function.
Its denominator is the product of linear forms dual to the vertices of
$\mathcal{P}$ 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 $S$.

###### Key words and phrases:

moments of a polytope, generating function

###### 2010 Mathematics Subject Classification:

Primary 44A60; Secondary 31B20

##  1\. Introduction

The initial motivation for the present paper came from proposed in [GLPR]
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 [Bri, La, MR1190788, Bar2], see [MR2455889, BR] for accessible
explanation. In [GLPR] 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 $\mathbb{R}^{d}$ is endowed with a
fixed coordinate system $(x_{1},...,x_{d})$, orthonormal with respect to the
standard scalar product $\langle\cdot,\cdot\rangle$. Let $\mu$ be a finite
complex-valued Borel measure in $\mathbb{R}^{d}$. (For standard measure-
theoretic notions we follow [MR924157].) Given a multiindex
$I=(i_{1},\dots,i_{d})$, let $\mathbf{x}^{I}$ be the shorthand of the monomial
$x_{1}^{i_{1}}\dots x_{d}^{i_{d}}$ and $|I|$ the shorthand for
$i_{1}+\dots+i_{d}$. For any multiindex $I$, define the moment $m_{I}(\mu)$ of
$\mu$ as

$m_{I}(\mu):=\int_{\mathbb{R}^{d}}x_{1}^{i_{1}}x_{2}^{i_{2}}...x_{d}^{i_{d}}d\mu(x_{1},x_{2},\ldots,x_{d})=\int_{\mathbb{R}^{d}}{\mathbf{x}}^{I}d\mu(\mathbf{x}).$
(1.1)

Define the normalized moment generating function
$F_{\mu}(\mathbf{u})=F_{\mu}(u_{1},\dots,u_{d})$ of $\mu$ by

$F_{\mu}(\mathbf{u}):=\sum_{I:=(i_{1},\dots,i_{d})\geq
0}\frac{(|I|+d)!}{i_{1}!\cdots
i_{d}!}m_{I}(\mu)\mathbf{u}^{I},\quad\text{where
$\mathbf{u}^{I}=u_{1}^{i_{1}}\dots u_{d}^{i_{d}}$.}$ (1.2)

Note that $F_{\mu}(\mathbf{u})$ admits the integral representation

$F_{\mu}(\mathbf{u})=d!\int_{\mathbb{R}^{d}}\frac{d\mu(\mathbf{x})}{(1-\langle\mathbf{x},\mathbf{u}\rangle)^{d+1}},$
(1.3)

which is a special case of a _Fantappiè transformation_. For details on the
latter, see e.g. [APS]*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 $\mu$ and any degree $\delta$
homogeneous $d$-variate polynomial $\rho$, it is convenient to define the
(re)normalized moment generating function $F_{\mu}^{\rho}(\mathbf{u})$ for the
measure $\rho\mu$, where by definition,
$\int_{\mathbb{R}^{d}}fd(\rho\mu)=\int_{\mathbb{R}^{d}}f\rho d\mu$, in such a
way that it can be obtained from $F_{\mu}(u)$ by application of the
differential operator $\rho\left(\frac{\partial}{\partial\mathbf{u}}\right)$.
Namely, set

$F_{\mu}^{\rho}(\mathbf{u}):=\sum_{I:=(i_{1},...,i_{d})\geq
0}\frac{(|I|+d+\delta)!}{i_{1}!\cdots i_{d}!}m_{I}(\rho\mu)\mathbf{u}^{I}.$
(1.4)

Note that $F_{\mu}^{\rho}(\mathbf{u})\neq F_{\rho\mu}(\mathbf{u})$ for non-
constant $\rho$. However, they are also connected, by an explicit differential
operator as follows.

######  Theorem 1.

For any complex-valued finite measure $\mu$ and any homogeneous polynomial
$\rho$ of degree $\delta$,

$\displaystyle F_{\mu}^{\rho}(\mathbf{u})$
$\displaystyle=\prod_{\ell=d}^{d+\delta-1}\left(\sum_{k}u_{k}\frac{\partial}{\partial
u_{k}}+\ell\right)\circ F_{\rho\mu}(\mathbf{u})$ (1.5)
$\displaystyle=\rho\left(\frac{\partial}{\partial\mathbf{u}}\right)\circ
F_{\mu}(\mathbf{u})$ (1.6)
$\displaystyle=(d+\delta)!\int_{\mathbb{R}^{d}}\frac{\rho(\mathbf{x})d\mu(\mathbf{x})}{(1-\langle\mathbf{x},\mathbf{u}\rangle)^{d+\delta+1}}.$
(1.7)

Here and in what follows $\circ$ 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 $S\subset\mathbb{R}^{d}$ is called spanning if it is not
contained in any (affine) hyperplane in $\mathbb{R}^{d}$. (Obviously,
$\text{card}(S)\geq d+1$.) As usual, by a (compact, convex) _polytope_
$\mathcal{P}\subset\mathbb{R}^{d}$ we mean the convex hull of a finite
spanning set in $\mathbb{R}^{d}$. The set of vertices of a convex polytope
$\mathcal{P}$ is the inclusion-minimal finite set with convex hull
$\mathcal{P}$. A $d$-simplex in $\mathbb{R}^{d}$ 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 $\mathcal{P}$ let
$\mathcal{V}=(\mathbf{v}_{1},...,\mathbf{v}_{N})$ denote the set of its
vertices. Assume that $\mathcal{P}$ is simple, i.e. each
$\mathbf{v}\in\mathcal{V}$ has exactly $d$ incident edges
$\mathbf{v}\mathbf{v}_{e_{1}}$, …, $\mathbf{v}\mathbf{v}_{e_{d}}$. Set
$w_{k}(\mathbf{v}):=\mathbf{v}_{e_{k}}-\mathbf{v}$, for $1\leq k\leq d$. The
non-negative real span $K_{\mathbf{v}}$ of $w_{1}(\mathbf{v})$,…,
$w_{d}(\mathbf{v})$ is called the tangent cone of $\mathcal{P}$ at
$\mathbf{v}$. For each $K_{\mathbf{v}}$ define $|\det
K_{\mathbf{v}}|=|\det(w_{1}(\mathbf{v}),\ldots,w_{d}(\mathbf{v}))|$ to be the
volume of the parallelepiped formed by
$w_{1}(\mathbf{v}),\ldots,w_{d}(\mathbf{v})$.

Given a bounded domain $\Omega\subset\mathbb{R}^{d}$, we call the measure

$\mu_{\Omega}=\chi_{\Omega}dx_{1}dx_{2}\ldots dx_{d},$

where $\chi_{\Omega}$ is the characteristic function of $\Omega$, the standard
measure of $\Omega$.

For a simple convex polytope $\mathcal{P}$ we have the following explicit
representation of $F_{\mu_{\mathcal{P}}}(\mathbf{u}).$

######  Theorem 2.

For an arbitrary simple convex polytope $\mathcal{P}$,

$\displaystyle F_{\mathcal{P}}(\mathbf{u}):=F_{\mu_{\mathcal{P}}}(\mathbf{u})$
$\displaystyle=(-1)^{d}\sum_{\mathbf{v}\in\mathcal{V}}\frac{\langle\mathbf{v},\mathbf{u}\rangle^{d}|\det
K_{\mathbf{v}}|}{\prod\limits_{j=1}^{d}\langle
w_{j}(\mathbf{v}),\mathbf{u}\rangle}\cdot\frac{1}{1-\langle\mathbf{v},\mathbf{u}\rangle}$
(1.8) $\displaystyle=(-1)^{d}\sum_{\mathbf{v}\in\mathcal{V}}\frac{|\det
K_{\mathbf{v}}|}{\prod\limits_{j=1}^{d}\langle
w_{j}(\mathbf{v}),\mathbf{u}\rangle}\cdot\frac{1}{1-\langle\mathbf{v},\mathbf{u}\rangle}.$
(1.9)

######  Remark 1.

Instead of the explicit choice of $w_{k}(\mathbf{v})$ for
$\mathbf{v}\in\mathcal{V}$ made above, we can take any fixed set of non-zero
vectors $w_{1}(\mathbf{v}),\ldots,w_{d}(\mathbf{v})$, spanning the tangent
cone of $\mathbf{v}$ in $\mathcal{P}$. This does not affect the validity of
(1.8) and (1.9).

Theorem 2 implies

######  Corollary 3.

Let $\Delta=\mathrm{conv}(\mathcal{V})\subset\mathbb{R}^{d}$ be an arbitrary
$d$-simplex. Then

$F_{\Delta}(\mathbf{u})=\frac{d!\mathrm{Vol}(\Delta)}{\prod\limits_{\mathbf{v}\in\mathcal{V}}(1-\langle\mathbf{v},\mathbf{u}\rangle)}.$
(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 [APS]*Section 3.5 and in
particular [APS]*Corollary 3.5.6.

A variation of (1.10) also appears in [BBDL], in the context of designing an
efficient procedure for integration of polynomials over simplices.

Notice that an arbitrary convex polytope $\mathcal{P}$ admits a triangulation
which only uses the existing vertices of $\mathcal{P}$, see e.g. [BR]*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 $F_{\mathcal{P}}^{\rho}(\mathbf{u})$
of any convex polytope $\mathcal{P}$ with respect to any homogeneous
polynomial density function $\rho$ of degree $\delta$ is a rational function
with denominator dividing

$\prod_{\mathbf{v}\in\mathcal{V}}(1-\langle\mathbf{v},\mathbf{u}\rangle)^{\delta}.$

######  Example 1.

Let $\Delta$ be a triangle in $\mathbb{R}^{2}$ with vertices
$v_{1}=(1,1),\;v_{2}=(2,5)$ and $v_{3}=(3,2)$. Its normalized moment
generating function equals

$F_{\Delta}(u_{1},u_{2})=\frac{7}{(1-u_{1}-u_{2})(1-2u_{1}-5u_{2})(1-3u_{1}-2u_{2})}.$

Its Taylor expansion about the origin up to the terms of degree $7$ is given
by

$7+42u_{1}+56u_{2}+175u_{1}^{2}+455u_{1}u_{2}+329u_{2}^{2}+630u_{1}^{3}+2387u_{1}^{2}u_{2}+3367u_{1}u_{2}^{2}+1750u_{2}^{3}+2107u_{1}^{4}+10318u_{1}^{3}u_{2}\\\
+21217u_{1}^{2}u_{2}^{2}+21546u_{1}u_{2}^{3}+8967u_{2}^{4}+6762u_{1}^{5}+40082u_{1}^{4}u_{2}+106526u_{1}^{3}u_{2}^{2}+157976u_{1}^{2}u_{2}^{3}+128772u_{1}u_{2}^{4}\\\
+45276u_{2}^{5}+21175u_{1}^{6}+145845u_{1}^{5}u_{2}+468895u_{1}^{4}u_{2}^{2}+900123u_{1}^{3}u_{2}^{3}+10744451u_{1}^{2}u_{2}^{4}+741993u_{1}u_{2}^{5}+227269u_{2}^{6},$

which implies that

$m_{00}=\frac{7}{2},m_{10}=7,m_{01}=\frac{28}{3},m_{20}=\frac{175}{12},m_{11}=\frac{455}{24},m_{02}=\frac{329}{12},m_{30}=\frac{63}{2},m_{21}=\frac{2387}{60},\\\
m_{12}=\frac{3591}{20},m_{03}=\frac{175}{2},m_{40}=\frac{2107}{30},m_{31}=\frac{5159}{60},m_{22}=\frac{21217}{180},m_{13}=\frac{3591}{20},m_{04}=\frac{2989}{10},\\\
m_{50}={161},m_{41}=\frac{2863}{15},m_{32}=\frac{7609}{30},m_{23}=\frac{5642}{15},m_{14}=\frac{3066}{5},m_{05}=1078,m_{60}=\frac{3025}{8},\\\
m_{51}=\frac{6945}{16},m_{42}=\frac{13397}{24},m_{33}=\frac{128589}{160},m_{24}=\frac{153493}{120},m_{15}=\frac{35333}{16},m_{06}=\frac{32467}{8}.$

### 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. [Bro1] [BroSt] [GMP] [GGMPV] [GMV00] [GLPR]. 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 $\mathcal{P}\subset\mathbb{R}^{d}$ 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 $\mathcal{P}$ is the number
of connected components of the set $\mathcal{P}^{o}\subset\mathcal{P}$ of
interior points of $\mathcal{P}$. The closure of each connected component of
$\mathcal{P}^{o}$ is called a component of $\mathcal{P}$. A generalized
polytope with one component is called indecomposable.

######  Remark 3.

We say that a simplicial complex in $\mathbb{R}^{d}$ is pure if all its
maximal simplices have dimension $d$. Clearly any generalized polytope in
$\mathbb{R}^{d}$ 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 $\mathcal{P}\subset\mathbb{R}^{d}$ 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.

######  Definition 3.

Given a generalized polytope $\mathcal{P}\subset\mathbb{R}^{d}$ we call a
finite collection of open disjoint $d$-dimensional simplices in
$\mathbb{R}^{d}$ a dissection of $\mathcal{P}$ if the closure of their union
coincides with $\mathcal{P}$.

A wealth of material on dissections of polytopes can be found in [Pak08], see
also [MR2743368].

######  Definition 4.

Given a generalized polytope $\mathcal{P}\subset\mathbb{R}^{d}$ we call a
point $\mathbf{v}$ a vertex of $\mathcal{P}$, if $\mathbf{v}$ is a vertex of
(the closure of) some open simplex in every dissection of $\mathcal{P}$.

######  Definition 5.

Given a point $p\in\mathcal{P}$ of a generalized polytope $\mathcal{P}$ we
denote by the tangent cone $T_{p}(\mathcal{P})$ of $\mathcal{P}$ at $p$ the
set obtained as follows. For a sufficiently small $\epsilon>0$ set
$\mathcal{P}_{p}(\epsilon)=\mathcal{P}\cap B_{p}(\epsilon)$ where
$B_{p}(\epsilon)$ is the $\epsilon$-ball centered at $p$. Define
$T_{p}(\mathcal{P})$ as the set obtained by taking a ray though $p$ and every
point of $\mathcal{P}_{p}(\epsilon)$. In other words, $T_{p}(\mathcal{P})$ is
the cone with the apex at $p$ and the base $B_{p}(\epsilon)$. (Obviously,
$T_{p}(\mathcal{P})$ is independent of $\epsilon$ for a sufficiently small
$\epsilon>0$.)

######  Lemma 5.

A point $\mathbf{v}$ is a vertex of $\mathcal{P}$ if and only if
$T_{\mathbf{v}}(\mathcal{P})$ 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 $\mathcal{P}$ at $\mathbf{v}$ has a connected component with
no translation-invariant direction then $\mathbf{v}$ is a vertex.

We denote by $\mathrm{conv}(S)$ the convex hull of an arbitrary set
$S\subset\mathbb{R}^{d}$. The above lemma implies that any vertex of
$\mathrm{conv}(\mathcal{P})$ is a vertex of $\mathcal{P}$.

The following result extends Corollary 4 to the case of generalized polytopes.

######  Proposition 6.

For any generalized polytope $\mathcal{P}$ with the set of vertices
$\mathcal{V}(\mathcal{P})$, the denominator of its normalized moment
generating function $F^{\rho}_{\mathcal{P}}(\mathbf{u})$ with respect to a
homogeneous polynomial density function $\rho$ of degree $\delta$ divides

$\Phi_{\mathcal{P}}(\mathbf{u}):=\prod_{\mathbf{v}\in\mathcal{V}(\mathcal{P})}(1-\langle\mathbf{v},\mathbf{u}\rangle)^{\delta}.$

######  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 [Sch]. Absence of a dissection $\mathcal{T}$
which uses only its $6$ vertices can be established by observing that none of
the edges $AC$, $A^{\prime}B$, and $B^{\prime}C^{\prime}$ can appear in a
simplex of $\mathcal{T}$, yet any simplex on these $6$ vertices must contain
one of them. Therefore, Proposition 6 is not an immediate consequence of
Corollary 3.

$0$$C^{\prime}$$A$$C$$B$$B^{\prime}$$A^{\prime}$$C$$C^{\prime}$$A^{\prime}$$B$$B^{\prime}$$A$
Figure 1. Schönhardt polyhedron obtained from an octahedron (on the left) by
removing tetrahedra $[ABB^{\prime}C]$, $[AA^{\prime}B^{\prime}C^{\prime}]$,
and $[A^{\prime}BCC^{\prime}]$.

######  Remark 6.

For “generic” generalized polytopes $\mathcal{P}$ the denominator
$\Omega(\mathbf{u})$ of $F_{\mathcal{P}}(\mathbf{u})$ equals
$\Phi_{\mathcal{P}}(\mathbf{u})$, but for certain special polytopes the
denominator $\Omega(\mathbf{u})$ may be its proper divisor, as can be seen
from the following example. Let
$A=\\{0,a_{1},a_{2},a_{3}\\}\subset\mathbb{R}^{3}$ be a spanning set, and
$v\in\mathbb{R}^{3}$. Let $\mathcal{P}_{\pm}:=\mathrm{conv}(v\pm A)$ and
$\mathcal{P}:=\mathcal{P}_{+}\cup\mathcal{P}_{-}$. Then
$1-\langle\mathbf{u},v\rangle$ does not appear in $\Omega(\mathbf{u})$, as

$F_{\mathcal{P}}(\mathbf{u})=F_{\mathcal{P}_{+}}(\mathbf{u})+F_{\mathcal{P}_{-}}(\mathbf{u})=K\frac{\sum\limits_{1\leq
i<j\leq
3}\langle\mathbf{u},a_{i}\rangle\langle\mathbf{u},a_{j}\rangle+(1-\langle\mathbf{u},v\rangle)^{2}}{\prod\limits_{1\leq
i\leq
3}((1-\langle\mathbf{u},v\rangle)^{2}-\langle\mathbf{u},a_{i}\rangle^{2})},$

where $K\neq 0$ is a real constant.

Now we introduce several finite-dimensional linear spaces related to a given
finite spanning set $S\subset\mathbb{R}^{d}$. Let $\mathcal{P}(S)$ be the set
of all generalized polytopes $\mathcal{P}$ whose sets
$\mathcal{V}(\mathcal{P})$ of vertices are contained in $S$. For
$\mathcal{P}\in\mathcal{P}(S)$ we denote by $\mu_{\mathcal{P}}$ its standard
measure. (Obviously, $\mu_{\mathcal{P}}$ is supported on
$\mathcal{P}\subseteq\mathrm{conv}(S)$.)

Denote by $\mathfrak{M}(S)$ the linear space of all signed measures, i.e. the
linear span of all standard measures $\mu_{\mathcal{P}}$ for
$\mathcal{P}\in\mathcal{P}(S)$. Let
$\mathfrak{M}^{\Delta}(S)\subseteq\mathfrak{M}(S)$ be its subspace spanned by
$\mu_{\Delta}$, for $\Delta\in\mathcal{P}(S)$ a $d$-dimensional simplex. (The
space $\mathfrak{M}^{\Delta}(S)$ has earlier appeared in [AlGelZel], [Al1],
[Al2] in a somewhat different context.) We shall refer to elements of
$\mathfrak{M}(S)$ 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 $\mathcal{P}$
with set of vertices contained in $S$, its standard measure
$\mu_{\mathcal{P}}$ belongs to $\mathfrak{M}^{\Delta}(S)$. In other words,
$\mathfrak{M}(S)=\mathfrak{M}^{\Delta}(S)$.

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
$S\subset\mathbb{R}^{d},$ 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 $\mathcal{P}$ can be obtained as
the limit of the standard measures of a $1$-parameter family of generalized
polytopes $\mathcal{P}(t)$ with $\mathcal{P}(0)=\mathcal{P}$ such that for
$t\neq 0$ the vertices of $\mathcal{P}(t)$ are weakly non-degenerate, and each
vertex of $\mathcal{P}(t)$ tending to a vertex of $\mathcal{P}$ as $t\to 0$.
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 $\mathfrak{M}(S)$. In
particular, we are able to compute the corresponding dimensions111 Note that
presently we are not aware of a formula or a recipe for calculating the
dimension of $\mathfrak{M}^{\Delta}(S)$ without the assumption of the theorem.
(The manuscript [Al1] contains an algorithm constructing a basis of this
space.) . In more detail, let $\mathfrak{F}(S)$ (resp.
$\mathfrak{F}^{\Delta}(S)$) be the linear space of Fantappiè transformations
of signed measures in $\mathfrak{M}(S)$ (resp. $\mathfrak{M}^{\Delta}(S)$). In
other words, $\mathfrak{F}(S)$ (resp. $\mathfrak{F}^{\Delta}(S)$) is the space
of normalized moment generating functions of signed measures in
$\mathfrak{M}(S)$ (resp. $\mathfrak{M}^{\Delta}(S)$).

Since each compactly supported measure is uniquely determined by its complete
set of moments, the map

$F_{\mu}:\mathfrak{M}(S)\to\mathfrak{F}(S),$ (1.11)

induced by the Fantappiè transformation is a linear isomorphism, cf.
[APS]*Section 3.5.

Finally, given a spanning set
$S=\\{\mathbf{v}_{1},\ldots,\mathbf{v}_{N}\\}\subset\mathbb{R}^{d},$ denote by
$\mathfrak{Rat}(S)$ the linear space of all rational functions with the
denominator $\Phi_{S}(\mathbf{u})$ as in (1.12),

$\Phi_{S}(\mathbf{u})=\prod_{i=1}^{N}(1-\langle\mathbf{v}_{i},\mathbf{u}\rangle),$
(1.12)

and with the numerator an arbitrary real (inhomogeneous) polynomial of degree
at most $N-d-1$. Here the numerator and the denominator might have common
factors.

######  Proposition 9.

$\mathfrak{F}^{\Delta}(S)$ coincides with $\mathfrak{Rat}(S)$ if and only if
$S$ is strongly non-degenerate.

######  Corollary 10.

If $S$ is strongly non-degenerate then
$\mathfrak{M}^{\Delta}(S)=\mathfrak{M}(S)$.

Corollary 10 implies that for strongly non-degenerate $S$ the dimension of all
these linear spaces equals $\binom{N-1}{d}$. 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 $S\subset\mathbb{R}^{d}$,
$|S|=N$, find the unique polytopal measure in $\mathfrak{M}(S)$ with a given
set of all moments up to order $N-d-1$.

We start with the following simple observation.

######  Lemma 11.

Given an arbitrary spanning set $S\subset\mathbb{R}^{d}$, $|S|=N$, and an
arbitrary polynomial $T(\mathbf{u})$ of degree at most $N-d-1$ there exists a
unique rational function $R(\mathbf{u})=P(\mathbf{u})/\Phi_{S}(\mathbf{u})$
with Taylor polynomial of degree $N-d-1$ at the origin equal to
$T(\mathbf{u})$. Namely,
$P(\mathbf{u})=\left[T(\mathbf{u})\Phi_{S}(\mathbf{u})\right]_{N-d-1}$, where
$\left[\cdot\right]_{N-d-1}$ stands for the truncated polynomial with all
monomials up to degree $N-d-1$.

For $S=\\{\mathbf{v}_{1},\ldots,\mathbf{v}_{N}\\}\subset\mathbb{R}^{d}$
strongly non-degenerate, we give an explicit inversion formula determining the
densities of an unknown polytopal measure having a given set of moments up to
order $N-d-1$ on each simplex in a natural basis of
$\mathfrak{M}^{\Delta}(S)$. In view of Lemma 11 we can assume that we are
already given an arbitrary rational function
$R(\mathbf{u})=P(\mathbf{u})/\Phi_{S}(\mathbf{u})$, where $\deg
P(\mathbf{u})\leq N-d-1$, and we want to determine the densities of the
required signed measure from $\mathfrak{M}(S)$ in terms of numerator
$P(\mathbf{u})$.

From now on we shall choose the basis of $\mathfrak{M}^{\Delta}(S)$ consisting
of the standard measures of all simplices containing the last vertex
$\mathbf{v}_{N}$, see Lemma 15 below. Let
$\mathfrak{L}=\\{l_{1},l_{2},....,l_{N-1}\\}$ be the $(N-1)$-tuple of linear
forms corresponding to vertices
$\mathbf{v}_{1},\mathbf{v}_{2},\ldots,\mathbf{v}_{N-1}$, where
$l_{i}(\mathbf{u})=1-\langle\mathbf{v}_{i},\mathbf{u}\rangle$. Consider the
linear span $V_{\mathfrak{L}}$ of all possible products of the form
$l_{j_{1}}\cdot l_{j_{2}}\cdot...\cdot l_{j_{N-d-1}},\;1\leq
j_{1}<j_{2}<...<j_{N-d-1}$. There are $\binom{N-1}{d}$ such products, and each
of them is a polynomial of degree at most $N-d-1$. On the other hand, the
dimension of the space $Pol(N-d-1,d)$ of all (inhomogeneous) polynomials of
degree at most $N-d-1$ in $d$ variables equals $\binom{N-1}{d}$, as well.

Define the square matrix $Mat_{S}$ of size $\binom{N-1}{d}$ with entries being
coefficients of the above products of linear forms with respect to the
standard monomial basis in $Pol(N-d-1,d)$. We assume that $Mat_{S}$ acts on
the space $V_{\mathfrak{L}}$ of column vectors.

######  Theorem 12.

For an arbitrary strongly non-degenerate spanning set
$S\subset\mathbb{R}^{d},$ $|S|=N$, the matrix $Mat_{S}$ is invertible.
Moreover, for a rational function
$R(\mathbf{u})=P(\mathbf{u})/\Phi_{S}(\mathbf{u})$, where $P(\mathbf{u})$ is
an arbitrary polynomial of degree $N-d-1$, there exists a unique measure
$\mu_{R}\in\mathfrak{M}(S)$ with Fantappiè transform $R(\mathbf{u})$. Namely,

$\mu_{R}=Mat_{S}^{-1}(P(\mathbf{u})).$ (1.13)

######  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
$Mat_{S}^{-1}$ 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 $\mu$ in $\mathbb{R}^{d}$ and a $d$-dimensional simplex
$\Delta\subset\mathbb{R}^{d}$ we define the weight $w_{\Delta}$ of $\Delta$ by
the formula:

$w_{\Delta}=d!\int_{\Delta}d\mu.$ (1.14)

In other words, the density $d_{\Delta}$ of the measure in question which
should be placed at $\Delta$ equals

$d_{\Delta}=\frac{w_{\Delta}}{d!\mathrm{Vol}(\Delta)}.$

We finish the introduction by explicitly solving the above inverse problem for
a concrete $5$-tuple of points in $\mathbb{R}^{2}$.

######  Example 2.

Set
$S=\\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3},\mathbf{v}_{4},\mathbf{v}_{5}\\}$
where
$\mathbf{v}_{1}=(1,0),\mathbf{v}_{2}=(2,1),\mathbf{v}_{3}=(1,2),\mathbf{v}_{4}=(0,1),\mathbf{v}_{5}=(0,0)$.
The corresponding set $\mathfrak{L}=\\{l_{1},l_{2},l_{3},l_{4}\\}$ of linear
forms is given by
$l_{1}=1-u_{1},\,l_{2}=1-2u_{1}-u_{2},\,l_{3}=1-u_{1}-2u_{2},\,l_{4}=1-u_{2}$.
Additionally, $l_{5}=1$. We are considering the basis of
$\mathfrak{M}^{\Delta}(S)$ consisting of (the standard measures of) 6
triangles containing $\mathbf{v}_{5}$. Therefore we need 6 quadratic forms
obtained as pairwise products $l_{i}l_{j},\,1\leq i<j\leq 4$. We get

$\begin{cases}l_{1}l_{2}=1-3u_{1}-u_{2}+2u_{1}^{2}+u_{1}u_{2}\\\
l_{1}l_{3}=1-2u_{1}-2u_{2}+u_{1}^{2}+2u_{1}u_{2}\\\
l_{1}l_{4}=1-u_{1}-u_{2}+u_{1}u_{2}\\\
l_{2}l_{3}=1-3u_{1}-3u_{2}+2u_{1}^{2}+5u_{1}u_{2}+2u_{2}\\\
l_{2}l_{4}=1-2u_{1}-2u_{2}+2u_{1}u_{2}+u_{2}^{2}\\\
l_{3}l_{4}=1-u_{1}-3u_{2}+u_{1}u_{2}+2u_{2}^{2}.\\\ \end{cases}$

Notice that $l_{1}l_{2}$ corresponds to triangle $\Delta_{345}$, $l_{1}l_{3}$
to $\Delta_{245}$, $l_{1}l_{3}$ to $\Delta_{245}$, $l_{1}l_{4}$ to
$\Delta_{234}$, $l_{2}l_{3}$ to $\Delta_{145}$, $l_{2}l_{4}$ to
$\Delta_{135}$, and $l_{3}l_{4}$ to $\Delta_{125}$. Ordering monomials
spanning the space $Pol(2,2)$ as
$(1,u_{1},u_{2},u_{1}^{2},u_{1}u_{2},u_{2}^{2})$ we get the $6\times 6$-matrix
$Mat_{S}$ and its inverse $Mat_{S}^{-1}$ as follows

$\blockarray{rrrrrrr}\block{cc\BAmulticolumn{4}{c}c}&Mat_{S}=\\\
&l_{1}l_{2}l_{1}l_{3}l_{1}l_{4}l_{2}l_{3}l_{2}l_{4}l_{3}l_{4}\\\
\block{r(rrrrrr)}1111111\\\ u_{1}-3-2-1-3-2-1\\\ u_{2}-1-2-1-3-2-3\\\
u_{1}^{2}210200\\\ u_{1}u_{2}121521\\\ u_{2}^{2}000212\\\
\quad\blockarray{rrrrrrr}\block{cc\BAmulticolumn{4}{c}c}&4Mat_{S}^{-1}=\\\
&1u_{1}u_{2}u_{1}^{2}u_{1}u_{2}u_{2}^{2}\\\
\block{r(rrrrrr)}l_{1}l_{2}1-1111-1\\\ l_{1}l_{3}-40-400-4\\\
l_{1}l_{4}933111\\\ l_{2}l_{3}111111\\\ l_{2}l_{4}-4-40-400\\\
l_{3}l_{4}11-11-11\\\ $

(For TeXnical reasons we give $4Mat_{S}^{-1}$ above.) Thus, given an arbitrary
rational function $R(u_{1},u_{2})=P(u_{1},u_{2})/\Phi_{S}(u_{1},u_{2})$ where
$P(u_{1},u_{2})=a_{00}+a_{1,0}u_{1}+a_{0,1}u_{2}+a_{2,0}u_{1}^{2}+a_{11}u_{1}u_{2}+a_{02}u_{2}^{2}$
is a polynomial of degree at most $2$ and
$\Phi_{S}(u_{1},u_{2})=l_{1}l_{2}l_{3}l_{4}l_{5}$, we get

$\begin{cases}w_{345}=\frac{1}{4}(a_{00}-a_{10}+a_{01}+a_{20}-a_{11}+a_{02})\\\
w_{245}=-a_{00}-a_{01}-a_{02}\\\
w_{235}=\frac{1}{4}(9a_{00}+3a_{01}+3a_{10}+a_{20}+a_{11}+a_{02})\\\
w_{145}=\frac{1}{4}(a_{00}+a_{01}+a_{10}+a_{20}+a_{11}+a_{02})\\\
w_{135}=-a_{00}-a_{10}-a_{20}\\\
w_{125}=\frac{1}{4}(a_{00}+a_{10}-a_{01}+a_{20}-a_{11}+a_{02}),\\\
\end{cases}$

where $w_{ijk}$ is the weight of the signed measure to be placed on
$\Delta_{ijk}$, see (1.14).

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 $F_{\mu}(\mathbf{u})$ satisfies the
relation

$F_{\mu}(\mathbf{u})=1\frac{2!}{0!0!}+2\frac{3!}{1!0!}u_{1}+3\frac{3!}{0!1!}u_{2}+4\frac{4!}{2!0!}u_{1}^{2}+5\frac{4!}{1!1!}u_{1}u_{2}+6\frac{4!}{0!2!}u_{2}^{2}+\dots=\frac{P(u_{1},u_{2})}{l_{1}l_{2}l_{3}l_{4}l_{5}},$

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

$P(u_{1},u_{2})=2+4u_{1}+10u_{2}+10u_{1}^{2}+24u_{1}u_{2}+10u_{2}^{2},$

i.e. $a_{00}=2,a_{10}=4,a_{01}=10,a_{20}=10,a_{11}=24,a_{02}=10$. Thus
$w_{345}=1,w_{245}=-22,w_{235}=26,w_{145}=15,w_{135}=-16,w_{125}=-2$. The
areas of the corresponding triangles are equal to:
$Area{(\Delta_{345})}=\frac{1}{2};Area({\Delta_{245}})=1;Area{(\Delta_{235})}=\frac{3}{2};Area{(\Delta_{145})}=\frac{1}{2};Area{(\Delta_{135})}=1;Area{(\Delta_{125})}=\frac{1}{2}.$
This implies that the densities of the measure of the corresponding triangles
are equal to
$d_{345}=1,d_{245}=-11,d_{235}=\frac{26}{3},d_{145}=15,d_{135}=-8,d_{125}=-2$.
To obtain the final densities in the convex hull $\mathrm{conv}(S)$ of $S$ one
has to decompose $\mathrm{conv}(S)$ into domains obtained by removing from
$\mathrm{conv}(S)$ the set of all hyperplanes spanned by vertices in $S$. For
each such domain we should add up the densities of all basic simplices
containing this domain. The resulting measure is shown in Fig. 2.

$\mathbf{v}_{5}=(0,0)$$\mathbf{v}_{1}=(1,0)$$\mathbf{v}_{2}=(2,1)$$\mathbf{v}_{3}=(1,2)$$\mathbf{v}_{4}=(0,1)$$5$$-10$$-2$$\frac{26}{3}$$\frac{31}{3}$$-\frac{7}{3}$$\frac{2}{3}$$1$$\frac{14}{3}$$5$$-10$
Figure 2. Final measure in Example 2.

######  Remark 9.

Domains into which the convex hull $\mathrm{conv}(S)$ is cut by the
hyperplanes spanned by $S$ were introduced in [AlGelZel] where they were
called chambers. The incidence matrix of the simplices spanned by $S$ and
those chambers was studied in some detail in [Al1],[Al2]. 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 [APS] in September 2011.

##  2\. Proving results on convex polytopes

Following Brion-Lawrence-Khovanskii-Pukhlikov-Barvinok, see [Bar2, BR, GLPR,
La], we define for each vector $\mathbf{z}\in\mathbb{R}^{d}$ the $j$-th
_axial_ moment $\mu_{j}(\mathbf{z})$ of a simple convex polytope $\mathcal{P}$
with respect to $\mathbf{z}$ as

$\mu_{j}(\mathbf{z})=\int_{\mathcal{P}}\langle\mathbf{x},\mathbf{z}\rangle^{j}d\mathbf{x}.$

We will use the following important statement, cf. e.g. [BR]*Theorem 10.5.

######  Theorem 13.

The moment $\mu_{j}(\mathbf{z})$ satisfies

$\mu_{j}(\mathbf{z})=\frac{(-1)^{d}j!}{(j+d)!}\sum_{\mathbf{v}\in\mathcal{V}}\langle\mathbf{v},\mathbf{z}\rangle^{j+d}D_{\mathbf{v}}(\mathbf{z}),$
(2.1)

where $D_{\mathbf{v}}(\mathbf{z}):=\frac{|\det
K_{\mathbf{v}}|}{\prod_{j=1}^{d}\langle w_{j}(\mathbf{v}),\mathbf{z}\rangle}$,
and $\mathbf{z}$ is an arbitrary vector for which the products
$\prod_{j=1}^{d}\langle w_{j}(\mathbf{v}),\mathbf{z}\rangle$,
$\mathbf{v}\in\mathcal{V}$, do not vanish. Moreover, the following identities
hold:

$\sum_{\mathbf{v}\in\mathcal{V}}\langle\mathbf{v},\mathbf{z}\rangle^{j}D_{\mathbf{v}}(\mathbf{z})=0,\quad\text{for
$0\leq j\leq d-1$.}$ (2.2)

###### Proof of Theorem 2.

To prove (1.8), consider the generating function

$\Psi_{\mathbf{z}}(u)=\sum_{j=0}^{\infty}\frac{(j+d)!}{j!}\mu_{j}(\mathbf{z})u^{j},$

where $u\in\mathbb{R}$. Formula (2.1) implies that $\Psi_{\mathbf{z}}(u)$ is
rational. Indeed,

$\Psi_{\mathbf{z}}(u)=\sum_{j=0}^{\infty}(-1)^{d}\sum_{\mathbf{v}\in\mathcal{V}}\langle\mathbf{v},\mathbf{z}\rangle^{j+d}\frac{|\det
K_{\mathbf{v}}|u^{j}}{\prod_{k=1}^{d}\langle
w_{k}(\mathbf{v}),\mathbf{z}\rangle}=\\\
=(-1)^{d}\sum_{\mathbf{v}\in\mathcal{V}}\frac{\langle\mathbf{v},\mathbf{z}\rangle^{d}|\det
K_{\mathbf{v}}|}{\prod_{k=1}^{d}\langle
w_{k}(\mathbf{v}),\mathbf{z}\rangle}\sum_{j=0}^{\infty}\langle\mathbf{v},\mathbf{z}\rangle^{j}u^{j}=\sum_{\mathbf{v}\in\mathcal{V}}\frac{\langle\mathbf{v},\mathbf{u}\rangle^{d}|\det
K_{\mathbf{v}}|}{\prod_{k=1}^{d}\langle
w_{k}(\mathbf{v}),\mathbf{u}\rangle}\cdot\frac{(-1)^{d}}{1-\langle\mathbf{v},\mathbf{u}\rangle},$

where $\mathbf{u}=u\mathbf{z}$. On the other hand, using the multinomial
coefficients $\binom{|J|}{J}=\frac{|J|!}{j_{1}!\dots j_{d}!}$ of multiindices
$J=(j_{1},\dots,j_{d})\vdash|J|$, one gets

$\int_{\mathcal{P}}\langle\mathbf{x},\mathbf{z}\rangle^{j}d\mathbf{x}=\int_{\mathcal{P}}\left(\sum_{i=1}^{d}x_{i}z_{i}\right)^{j}d\mathbf{x}=\sum_{J\vdash
j}\binom{j}{J}\mathbf{z}^{J}\int_{\mathcal{P}}\mathbf{x}^{J}d\mathbf{x}=\sum_{J\vdash
j}\binom{j}{J}\mathbf{z}^{J}m_{{}_{J}}(\mathcal{P}),$

where $m_{{}_{J}}(\mathcal{P})=m_{{}_{J}}(\mu_{\mathcal{P}})$. Therefore,

$F_{\mathcal{P}}(u\mathbf{z}):=F_{\mathcal{P}}(uz_{1},...,uz_{d}):=\sum_{j=0}^{\infty}\sum_{(j_{1},...,j_{d})\vdash
j}\frac{(j+d)!}{j_{1}!\dots
j_{d}!}m_{j_{1},\dotsc,j_{d}}(uz_{1})^{j_{1}}\cdots(uz_{d})^{j_{d}}\\\
=\sum_{j=0}^{\infty}\frac{(j+d)!}{j!}\left[\sum_{J:=(j_{1},...,j_{d})\vdash
j}\binom{j}{J}m_{{}_{J}}(\mathcal{P})\mathbf{z}^{J}\right]u^{j}=\sum_{j=0}^{\infty}\frac{(j+d)!}{j!}\mu_{j}(\mathbf{z})u^{j}=\Psi_{\mathbf{z}}(u),$

and (1.8) follows.

In view of relations (2.2) the right-hand side of (1.8) can be rewritten as
(1.9). Indeed, writing
$(1-\langle\mathbf{v},\mathbf{u}\rangle)^{-1}=\sum_{j=0}^{\infty}\langle\mathbf{v},\mathbf{u}\rangle^{j}$
and expanding (1.8) with respect to $j$th powers of
$\langle\mathbf{v},\mathbf{u}\rangle$, we see that (2.2) implies that for
$j<d$ the sum of all terms $\langle\mathbf{v},\mathbf{u}\rangle^{j}$ vanishes.
∎

###### Proof of Corollary 3.

Let $\mathcal{V}=(\mathbf{v}_{0},\mathbf{v}_{1},\ldots,\mathbf{v}_{d})$. Then
for each $j\neq i$ we have
$w_{j}(\mathbf{v}_{i})=\mathbf{v}_{j}-\mathbf{v}_{i}$. Hence $|\det
K_{\mathbf{v}_{i}}|$ does not depend upon $i$ and equals
$d!\mathrm{Vol}(\Delta)$. The right-hand side of (1.8) becomes

$(-1)^{d}d!\mathrm{Vol}(\Delta)\sum_{i=0}^{d}\frac{\langle\mathbf{v}_{i},\mathbf{u}\rangle^{d}}{\prod_{j=1}^{d}\langle
w_{j}(\mathbf{v}_{i}),\mathbf{u}\rangle}\cdot\frac{1}{1-\langle\mathbf{v}_{i},\mathbf{u}\rangle}=\\\
=(-1)^{d}d!\mathrm{Vol}(\Delta)\sum_{i=0}^{d}\frac{\langle\mathbf{v}_{i},\mathbf{u}\rangle^{d}}{\prod_{j\neq
i}\langle\mathbf{v}_{j}-\mathbf{v}_{i},\mathbf{u}\rangle}\cdot\frac{1}{1-\langle\mathbf{v}_{i},\mathbf{u}\rangle}=\\\
=(-1)^{d}d!\mathrm{Vol}(\Delta)\sum_{i=0}^{d}\frac{\zeta_{i}^{d}}{\prod_{j\neq
i}(\zeta_{j}-\zeta_{i})}\cdot\frac{1}{1-\zeta_{i}},$

where $\zeta_{i}=\langle\mathbf{v}_{i},\mathbf{u}\rangle$. Computing the
common denominator of the latter, we obtain

$F_{\Delta}(\mathbf{u})=\frac{(-1)^{d}d!\mathrm{Vol}(\Delta)}{\prod_{i=0}^{d}(1-\zeta_{i})}\frac{\sum\limits_{i=0}^{d}\left[\prod\limits_{k>l,k\neq
i\neq l}(\zeta_{k}-\zeta_{l})\prod\limits_{j\neq
i}(1-\zeta_{j})\right](-1)^{i}\zeta_{i}^{d}}{\prod_{s>t}(\zeta_{s}-\zeta_{t})}.$

It is convenient to introduce one more linear form $\zeta_{d+1}:=1$, so that
the last expression reads as

$F_{\Delta}(\mathbf{u})=(-1)^{d}d!\mathrm{Vol}(\Delta)\frac{\sum\limits_{i=0}^{d}\left[\prod\limits_{\begin{subarray}{c}d+1\geq
k>l\geq 0,\\\ k\neq i\neq
l\end{subarray}}(\zeta_{k}-\zeta_{l})\right](-1)^{i}\zeta_{i}^{d}}{\prod\limits_{d+1\geq
s>t\geq 0}(\zeta_{s}-\zeta_{t})}.$ (2.3)

To complete the proof we notice that

$0=\det\begin{pmatrix}1&1&\ldots&1&1\\\
\zeta_{0}&\zeta_{1}&\ldots&\zeta_{d}&1\\\
\zeta_{0}^{2}&\zeta_{1}^{2}&\ldots&\zeta_{d}^{2}&1\\\ \ldots\\\
\zeta_{0}^{d}&\zeta_{1}^{d}&\ldots&\zeta_{d}^{d}&1\\\
\zeta_{0}^{d}&\zeta_{1}^{d}&\ldots&\zeta_{d}^{d}&1\\\
\end{pmatrix}=1\cdot\det\begin{pmatrix}1&1&\ldots&1\\\
\zeta_{0}&\zeta_{1}&\ldots&\zeta_{d}\\\
\zeta_{0}^{2}&\zeta_{1}^{2}&\ldots&\zeta_{d}^{2}\\\ \ldots\\\
\zeta_{0}^{d}&\zeta_{1}^{d}&\ldots&\zeta_{d}^{d}\\\ \end{pmatrix}+\\\
+(-1)^{d+1}\sum_{i=0}^{d}\zeta_{i}^{d}(-1)^{i}\cdot\det\begin{pmatrix}1&\ldots&1&1&\ldots&1\\\
\zeta_{0}&\ldots&\zeta_{i-1}&\zeta_{i+1}&\ldots&\zeta_{d+1}\\\
\zeta_{0}^{2}&\ldots&\zeta_{i-1}^{2}&\zeta_{i+1}^{2}&\ldots&\zeta_{d+1}^{2}\\\
\ldots\\\
\zeta_{0}^{d}&\ldots&\zeta_{i-1}^{d}&\zeta_{i+1}^{d}&\ldots&\zeta_{d+1}^{d}\\\
\end{pmatrix}=\\\ =\prod_{d\geq k>l\geq
0}(\zeta_{k}-\zeta_{l})+(-1)^{d+1}\sum_{i=0}^{d}\zeta_{i}^{d}(-1)^{i}\cdot\left[\prod\limits_{\begin{subarray}{c}d+1\geq
k>l\geq 0,\\\ k\neq i\neq l\end{subarray}}(\zeta_{k}-\zeta_{l})\right].$

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

$\sum_{i=0}^{d}\zeta_{i}^{d}(-1)^{i}\cdot\left[\prod\limits_{\begin{subarray}{c}d+1\geq
k>l\geq 0,\\\ k\neq i\neq
l\end{subarray}}(\zeta_{k}-\zeta_{l})\right]=(-1)^{d}\prod_{d\geq k>l\geq
0}(\zeta_{k}-\zeta_{l}).$

Now we plug this formula into (2.3) and get

$F_{\Delta}(\mathbf{u})=d!\mathrm{Vol}(\Delta)\frac{1}{\prod\limits_{t=0}^{d}(1-\zeta_{t})}=\frac{d!\mathrm{Vol}(\Delta)}{\prod\limits_{\mathbf{v}\in\mathcal{V}}(1-\langle\mathbf{v},\mathbf{u}\rangle)}.\qed$

######  Lemma 14.

Let $\mathbf{u}=(u_{1},\dots,u_{d})$ and $\mathbf{x}=(x_{1},\dots,x_{d})$ be
formal variables, and $\ell\in\mathbb{R}$ . Then

$\left(\sum_{k}u_{k}\frac{\partial}{\partial
u_{k}}+\ell\right)\circ(1-\langle\mathbf{x},\mathbf{u}\rangle)^{-\ell}=\ell(1-\langle\mathbf{x},\mathbf{u}\rangle)^{-\ell-1}.$
(2.4)

###### Proof.

Note that $u_{k}\frac{\partial}{\partial
u_{k}}\circ(1-\langle\mathbf{x},\mathbf{u}\rangle)^{-\ell}=x_{k}u_{k}\ell(1-\langle\mathbf{x},\mathbf{u}\rangle)^{-\ell-1}$.
Thus

$\left(\sum_{k}u_{k}\frac{\partial}{\partial
u_{k}}+\ell\right)\circ(1-\langle\mathbf{x},\mathbf{u}\rangle)^{-\ell}=\ell(1-\langle\mathbf{x},\mathbf{u}\rangle)^{-\ell-1}\langle\mathbf{x},\mathbf{u}\rangle+\ell(1-\langle\mathbf{x},\mathbf{u}\rangle)^{-\ell}\\\
=\ell(1-\langle\mathbf{x},\mathbf{u}\rangle)^{-\ell-1}.\qed$

###### Proof of (1.3).

For a $d$-variate polynomial $g(\mathbf{z})$ we denote by
$g\left(\mathbf{u}\frac{\partial}{\partial\mathbf{u}}\right)$ the differential
operator $g\left(u_{1}\frac{\partial}{\partial
u_{1}},\dots,u_{d}\frac{\partial}{\partial u_{d}}\right)$. We use the identity

$g\left(\mathbf{u}\frac{\partial}{\partial\mathbf{u}}\right)\circ\sum_{I}a_{{}_{I}}\mathbf{x}^{I}\mathbf{u}^{I}=\sum_{I\geq
0}g(I)a_{{}_{I}}\mathbf{x}^{I}\mathbf{u}^{I},$ (2.5)

which holds for any formal $d$-variate power series
$\sum_{I}a_{I}\mathbf{x}^{I}\mathbf{u}^{I}$ and any $d$-variate polynomial
$g(\mathbf{z})$. (It can be easily verified for monomial $g(\mathbf{z})$ and
then extended by linearity.) Setting
$h(\mathbf{z}):=\prod_{\ell=1}^{d}\left(\sum_{k=1}^{d}z_{k}+\ell\right),$
notice that $h(I)=(|I|+1)(|I|+2)\cdots(|I|+d)$. Now using (2.5) together with
the obvious identity:

$(1-\langle\mathbf{x},\mathbf{u}\rangle)^{-1}=\sum_{I\geq
0}\binom{|I|}{I}\mathbf{x}^{I}\mathbf{u}^{I},$

one obtains

$F_{\mu}(\mathbf{u}):=\sum_{I\geq
0}\binom{|I|+d}{I}m_{I}(\mu)\mathbf{u}^{I}=\sum_{I\geq
0}h(I)\binom{|I|}{I}m_{I}(\mu)\mathbf{u}^{I}\\\
=\int_{\mathbb{R}^{d}}\sum_{I\geq
0}h(I)\binom{|I|}{I}\mathbf{x}^{I}\mathbf{u}^{I}d\mu(\mathbf{x})=\int\limits_{\mathbb{R}^{d}}h\left(\mathbf{u}\frac{\partial}{\partial\mathbf{u}}\right)\circ\sum_{I\geq
0}\binom{|I|}{I}\mathbf{x}^{I}\mathbf{u}^{I}d\mu(\mathbf{x})\\\
=\int\limits_{\mathbb{R}^{d}}h\left(\mathbf{u}\frac{\partial}{\partial\mathbf{u}}\right)\circ\frac{d\mu(\mathbf{x})}{1-\langle\mathbf{x},\mathbf{u}\rangle}=\int\limits_{\mathbb{R}^{d}}\frac{d!\
d\mu(\mathbf{x})}{(1-\langle\mathbf{x},\mathbf{u}\rangle)^{d+1}},$

where in the final derivation we repeatedly made use of (2.4), for
$1\leq\ell\leq d.$ ∎

######  Remark 10.

Another point of view on (1.3) is that it is the result of the application of
the differential operator
$g\left(\mathbf{u}\frac{\partial}{\partial\mathbf{u}}\right)$ to the integral
transformation
$\int_{\mathbb{R}^{d}}\frac{d\mu(\mathbf{x})}{1-\langle\mathbf{x},\mathbf{u}\rangle}$
of the measure $\mu$ (also known as the Fantappiè transform of $\mu$); see
e.g. [MR2274973].

In [PS12] a similar idea was applied to the harmonic polygonal measures in the
plane.

###### Proof of Theorem 1.

Assume first that $\rho(x_{1},...,x_{d})={\bf x}^{K}=x_{1}^{k_{1}}\cdots
x_{d}^{k_{d}}$ is a monomial and consider

$\rho\left(\frac{\partial}{\partial{\mathbf{u}}}\right)\circ
F_{\mu}({\mathbf{u}})=\frac{\partial^{|{K}|}}{\partial
u_{1}^{k_{1}}\cdots\partial u_{d}^{k_{d}}}\circ\sum_{I=(i_{1},\dots,i_{d})\geq
0}\frac{(|I|+d)!}{i_{1}!\cdots i_{d}!}m_{I}(\mu)\mathbf{u}^{I}.$

One gets

$\frac{\partial^{|K|}}{\partial{\mathbf{u}}^{K}}\circ
F_{\mu}({\mathbf{u}})=\sum_{I=(i_{1},...,i_{d})\geq
0}\left(\frac{(|I|+|{K}|+d)!}{\prod\limits_{j=1}^{d}(i_{j}+k_{j})!}\prod_{j=1}^{d}\frac{(i_{j}+k_{j})!}{i_{j}!}\right)m_{I+K}(\mu)\mathbf{u}^{I}\\\
=\sum_{I=(i_{1},\dots,i_{d})\geq
0}\frac{(|I|+d+|{K}|)!}{\prod_{j=1}^{d}i_{j}!}m_{I}(\mathbf{x}^{K}\mu)\mathbf{u}^{I}.$

Observe that the normalizing coefficients of $m_{I}(\mathbf{x}^{K}\mu)$ in the
latter expression depend only on $I$ and $|K|$ but not on particular entries
of $K$. Therefore for an arbitrary homogeneous $\rho$ of degree $\delta$ one
gets by additivity

$\rho\left(\frac{\partial}{\partial{\mathbf{u}}}\right)\circ
F_{\mu}({\mathbf{u}})=\sum_{I=(i_{1},\dots,i_{d})\geq
0}\frac{(|I|+d+\delta)!}{\prod\limits_{j=1}^{d}i_{j}!}m_{I}(\rho\mu).$

This shows (1.6). Repeated application of (2.4), for $d+1\leq\ell\leq
d+\delta$, to the integral representation (1.3), respectively, to the
representation (1.2), of $F_{\rho\mu}(\mathbf{u})$ implies (1.7),
respectively, (1.5). ∎

##  3\. 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 $\mathbf{v}$ be a point in $\mathcal{P}$ which is not a vertex. Then there
is a dissection $\mathcal{T}$ of $\mathcal{P}$ such that $\mathbf{v}$ is not a
vertex of any simplex of $\mathcal{T}$. Let $U$ be the set of simplices
$S_{u}$ of $\mathcal{T}$ with closures containing $\mathbf{v}$. Take the
dissection of the tangent cone $T_{v}(\mathcal{P})$ into the tangent cones to
simplices from $U$, $T_{\mathbf{v}}(\mathcal{P})=\cup_{u\in
U}T_{\mathbf{v}}(S_{u})$. Clearly, every subcone $T_{v}(S_{u})$ contains a
translation-invariant direction (any direction parallel to the minimal face
containing $\mathbf{v}$).

Vice versa, to prove the converse implication, let us take a dissection of the
tangent cone $T_{\mathbf{v}}(\mathcal{P})$ into a disjoint union of convex
polytopal cones $Q_{1},\dots,Q_{k}$. By definition of the tangent cone and
since $\mathcal{P}$ can be represented as a finite union of simplices we
obtain that any sufficiently small neighborhood of $\mathbf{v}$ in the tangent
cone $T_{\mathbf{v}}(\mathcal{P})$ is a neighborhood of $\mathbf{v}$ in the
entire $\mathcal{P}$. Consider the parallelepiped $\text{Box}_{\varepsilon}$
centered at $\mathbf{v}$ that is $\varepsilon$ ball centered at $\mathbf{v}$,
in the $L_{1}$-norm. Note that each convex polytopal set
$Q_{i}\cap\text{Box}_{\varepsilon}$ can be decomposed into a union of
simplices that do not contain $\mathbf{v}$ as a vertex.

Further notice that the set $\mathcal{P}\setminus\text{Box}_{\varepsilon}$ can
be represented as a finite disjoint union of simplices, since
$\text{Box}_{\varepsilon}$ is the intersection of a finite number of half-
spaces and $\mathcal{P}$ is a disjoint union of simplices. Clearly, every
simplex in this union should not have $\mathbf{v}$ as a vertex. Now combining
the dissections of each $Q_{i}\cap\text{Box}_{\varepsilon}$ and
$\mathcal{P}\setminus\text{Box}_{\varepsilon}$ we obtain the required
dissection of $\mathcal{P}$. ∎

###### Proof of Proposition 6.

We begin by considering the case $\rho\equiv 1$. Let $\mathcal{T}$ be a
dissection of $\mathcal{P}$ with vertices $\mathcal{V}(\mathcal{T})$.
Corollary 3 implies that $F_{\mathcal{P}}(\mathbf{u})$ has a denominator
dividing
$g_{\mathcal{T}}(\mathbf{u})=\prod_{\mathbf{v}\in\mathcal{V}(\mathcal{T})}(1-\langle\mathbf{v},\mathbf{u}\rangle)$.
Take
$\mathbf{v}_{1}\in\mathcal{V}(\mathcal{T})\setminus\mathcal{V}(\mathcal{P})$.
Then there exists another dissection $\mathcal{T}^{\prime}$ such that
$\mathbf{v}_{1}\not\in\mathcal{V}(\mathcal{T}^{\prime})$. Expressing
$F_{\mathcal{P}}(\mathbf{u})$ as ratios of polynomials, we have

$F_{\mathcal{P}}(\mathbf{u})=\frac{f_{\mathcal{T}}(\mathbf{u})}{h_{\mathcal{T}}(\mathbf{u})(1-\langle\mathbf{v}_{1},\mathbf{u}\rangle)}=\frac{f_{\mathcal{T}^{\prime}}(\mathbf{u})}{g_{\mathcal{T}^{\prime}}(\mathbf{u})},\quad\text{where
}g_{\mathcal{T}}(\mathbf{u})={h_{\mathcal{T}}(\mathbf{u})(1-\langle\mathbf{v}_{1},\mathbf{u}\rangle)}.$

Here $g_{\mathcal{T}^{\prime}}$ is not divisible by
$1-\langle\mathbf{v}_{1},\mathbf{u}\rangle$, by the choice of
$\mathcal{T}^{\prime}$. Thus $f_{\mathcal{T}}$ is divisible by
$1-\langle\mathbf{v}_{1},\mathbf{u}\rangle$, and can be canceled out in the
expression for $F_{\mathcal{P}}(\mathbf{u})$.

The case of arbitrary homogeneous $\rho$ follows immediately by applying
Theorem 1 to the already covered case $\rho\equiv 1$. ∎

###### Proof of Proposition 9.

First we show that for an arbitrary finite spanning set
$S\subset\mathbb{R}^{d}$ the space $\mathfrak{M}^{\Delta}(S)$ has a basis of
$d$-dimensional simplices containing a fixed vertex $\mathbf{v}\in S$. In
particular, the set of all $d$-dimensional simplices containing $\mathbf{v}$
spans $\mathfrak{M}^{\Delta}(S)$ but is not necessarily a basis. Consequently,
their Fantappiè transformations spans $\mathfrak{F}^{\Delta}(S)$. The
following result is formulated as Theorem 4.2 of [Al1] and in a different form
in [AlGelZel]. (We omit the proof of this statement here.)

Given two points $p$ and $q$ and a set $M$ in $\mathbb{R}^{d}$, we say that
$q$ is visible from $p$ with respect to $M$ if the straight interval $pq$ is
disjoint from $M$.

######  Lemma 15.

Given a $d$-dimensional simplex $\sigma\subset\mathbb{R}^{d}$ denote by
$\mathcal{V}(\sigma)$ the set of vertices of $\sigma$. Let $\sigma^{0}$ be the
interior of $\sigma$. Let $p$ be any point in $\mathbb{R}^{d}$ and let $Q^{+}$
(resp. $Q^{-}$) be the set of all $(d-1)$-dimensional faces of $\sigma$ which
are visible (resp. not visible) from $p$ with respect to $\sigma^{0}$. Then
the standard measures of all $d$-dimensional simplices with vertices in
$\mathcal{V}(\sigma)\cup\\{p\\}$ satisfy

$\mu_{\sigma}=\sum_{\sigma_{i}\in Q^{+}}\mu_{\sigma_{i},p}-\sum_{\sigma_{i}\in
Q^{-}}\mu_{\sigma_{i},p},$

where $\mu_{\sigma_{i},p}$ is the standard measure of the $d$-dimensional
simplex spanned by the vertices of $\sigma_{i}$ and the point $p$.

######  Remark 11.

If $\sigma_{i,p}$ is a degenerate simplex, i.e. $p$ lies in the hyperplane
spanned by $\sigma_{i}$, we simply exclude the corresponding term
$\mu_{\sigma_{i,p}}$ from the above formula.

To prove Proposition 9, we need to show that $\mathfrak{F}^{\Delta}(S)$
coincides with $\mathfrak{Rat}(S)$ if and only if $S$ is strongly non-
degenerate. Indeed, $\mathfrak{F}^{\Delta}(S)\subseteq\mathfrak{Rat}(S)$ for
an arbitrary spanning $S$, by Proposition 6. The Fantappiè transform
$F_{\mu}:\mathfrak{M}^{\Delta}(S)\to\mathfrak{F}^{\Delta}(S)$ is a linear
isomorphism which implies that
$\dim\mathfrak{M}^{\Delta}(S)=\dim\mathfrak{F}^{\Delta}(S)$. By Lemma 15 the
space $\mathfrak{M}^{\Delta}(S)$ is spanned by the standard measures
$\mu_{\Delta}$ of the set $\mathcal{B}_{i}$ of all $d$-dimensional simplices
containing the fixed vertex $\mathbf{v}_{i}$. Let us fix the vertex
$\mathbf{v}_{{}_{N}}$ and consider the set $\mathcal{B}_{{}_{N}}$. For $S$
strongly non-degenerate the cardinality of $\mathcal{B}_{{}_{N}}$ equals
$\binom{N-1}{d}$.

Now we show that $\mathfrak{F}^{\Delta}(S)=\mathfrak{Rat}(S)$, where
$\mathfrak{Rat}(S)$ has the dimension $\binom{N-1}{d}$, as it is isomorphic to
the space $Pol(N-d-1,d)$ of all $d$-variate polynomials of degree at most
$N-d-1$. This would immediately imply that the standard measures of simplices
in $\mathcal{B}_{{}_{N}}$ are linearly independent.

######  Lemma 16.

If $S$ is strongly non-degenerate, then
$\mathfrak{F}^{\Delta}(S)=\mathfrak{Rat}(S)$.

###### Proof.

We recall that $\mathfrak{F}^{\Delta}(S)$ comprises all linear combinations of
the rational functions

$\frac{d!\mathrm{Vol}(\mathbf{v}_{i_{1}},\dots,\mathbf{v}_{i_{d}},\mathbf{v}_{{}_{N}})}{(1-\langle\mathbf{v}_{i_{1}},\mathbf{u}\rangle)\cdot\dots\cdot(1-\langle\mathbf{v}_{i_{d}},\mathbf{u}\rangle)\cdot(1-\langle\mathbf{v}_{{}_{N}},\mathbf{u}\rangle)}.$

For each term $1-\langle\mathbf{v}_{i},\mathbf{u}\rangle$ we consider a
(homogeneous) linear form
$l_{i}(u_{0},\mathbf{u})=u_{0}-\langle\mathbf{v}_{i},\mathbf{u}\rangle$ in
$d+1$ variables $u_{0},\dots,u_{d}$. Set $n=N-1$ where $n\geq d+1$. For the
$n$-tuple $\mathfrak{L}=\\{l_{1},l_{2},....,l_{n}\\}$ of linear
$(d+1)$-variate forms, let $V_{\mathfrak{L}}$ be the linear span of all
possible products of the form $l_{i_{1}}l_{i_{2}}\dots l_{i_{n-d}},\;1\leq
i_{1}<i_{2}<...<i_{n-d}\leq n$. Observe that $V_{\mathfrak{L}}$ is the space
of all numerators that one can obtain in $\mathfrak{F}^{\Delta}(S)$. We need
to show that $V_{\mathfrak{L}}$ contains $HPoly(n-d,d+1)$, the space of all
$d+1$-variate homogeneous polynomials of degree $n-d$. Recall that any
$d+1$-tuple of linear forms $l_{i_{1}},\dots,l_{i_{d+1}}$ is linearly
independent due to the strong degeneracy assumption. Thus we can express each
single variable $u_{0},\dots,u_{d}$ as a linear combination of these forms.
Since $V_{\mathfrak{L}}$ contains all products $l_{i_{1}}l_{i_{2}}\dots
l_{i_{n-d-1}}l_{j}$, where
$j\in\\{1,\dots,n\\}\setminus\\{i_{1},\dots,i_{n-d-1}\\}$, we conclude that
$V_{\mathfrak{L}}$ contains all homogeneous polynomials of the form

$l_{i_{1}}l_{i_{2}}\dots l_{i_{n-d-1}}u_{k},\quad\text{ for $0\leq k\leq d$.}$

From that we deduce that $V_{\mathfrak{L}}$ contains all homogeneous
polynomials of the form $l_{i_{1}}l_{i_{2}}\dots l_{i_{n-d-2}}u_{k}u_{j}$,
where $j,k\in[n]\setminus\\{i_{1},\dots,i_{n-d-1}\\}$. Continuing along the
same lines we derive by induction that $V_{\mathfrak{L}}$ contains
$HPoly(n-d,d+1)$. ∎

For an arbitrary spanning $S$ the cardinality of $\mathcal{B}_{{}_{N}}$ is at
most $\binom{N-1}{d}=\dim\mathfrak{Rat}(S)$. Furthermore, if $S$ is not
strongly non-degenerate the cardinality of $\mathcal{B}_{{}_{N}}$ is strictly
smaller than $\binom{N-1}{d}$, as there will be linear dependencies among the
standard measures on the simplices in $B_{{}_{N}}$. Therefore
$\dim\mathfrak{F}^{\Delta}(S)<\dim\mathfrak{Rat}(S)$. ∎

We define the square matrix $Mat_{\mathfrak{L}}$ of size $\binom{n}{d}$ with
entries being coefficients of the above products of linear forms w.r.t. the
standard monomial basis in $HPol(n-d,d+1)$.

######  Lemma 17.

The determinant of $Mat_{\mathfrak{L}}$ is proportional to the product of the
determinants of all $(d+1)$-tuples
$(l_{i_{1}},l_{i_{2}},\ldots,l_{i_{d+1}}),\;i_{1}<i_{2}<\ldots<i_{d+1}$. (By
the determinant of a $(d+1)$-tuple of vectors in $\mathbb{R}^{d+1}$ with a
fixed basis we mean the determinant of the matrix formed by the coordinates of
these vectors in a chosen basis.)

###### Proof.

Indeed, $\det(Mat_{\mathfrak{L}})$ is a form of degree $(d+1)\binom{n}{d}$ in
the coefficients of the linear forms $l_{1},\ldots,l_{n}$. Thus the product
$\prod_{i_{1},\ldots,{i_{d+1}}}\det(l_{i_{1}},l_{i_{2}},\ldots,l_{i_{d+1}})$
has the same degree as $\det(Mat_{\mathfrak{L}})$. Therefore it suffices to
show that $\det(Mat_{\mathfrak{L}})$ vanishes as soon as some of
$\det(l_{i_{1}},l_{i_{2}},\ldots,l_{i_{d+1}})$ vanishes. (Observe that all
polynomials $\det(Mat_{\mathfrak{L}})$ are coprime.) Without loss of
generality assume that $l_{1}$ is a linear combination of
$l_{2},\ldots,l_{d+1}$. But then the column of $Mat_{\mathfrak{L}}$
corresponding to the $(n-d)$-tuple $(1,d+2,d+3,\ldots,n)$ will be a linear
combination of those corresponding to $(2,d+2,d+3,\ldots,n),$
…$(d+1,d+2,d+3,\ldots,n)$. ∎

###### Proof of Corollary 10.

As we mentioned above, $\mathfrak{M}(S)$ is isomorphic to $\mathfrak{F}(S)$
and, analogously, $\mathfrak{M}^{\Delta}(S)$ is isomorphic to
$\mathfrak{F}^{\Delta}(S)$. Thus, if we prove the equality
$\mathfrak{F}(S)=\mathfrak{F}^{\Delta}(S)$ then we get
$\mathfrak{M}(S)=\mathfrak{M}^{\Delta}(S)$. By Lemma 16, the space
$\mathfrak{F}^{\Delta}(S)$ coincides with the linear space of all rational
functions with the numerator an arbitrary polynomial of degree at most $N-d-1$
and the denominator $\Phi_{S}(\mathbf{u})$ equal to the product of all linear
forms dual to all vertices in $S$. By Proposition 6 an arbitrary function in
$\mathfrak{F}(S)$ is a rational function with denominator of desired form and
numerator of degree at most $N-d-1$, for obvious reasons—take an arbitrary
dissection and sum over its simplices. Since all such functions are already in
$\mathfrak{F}^{\Delta}(S)$ we are done. ∎

###### Proof of Theorem 12.

Given a strongly non-degenerate set
$S=\\{\mathbf{v}_{1},\ldots,\mathbf{v}_{N-1},\mathbf{v}_{N}\\}$ and the
Fantappiè transform $R(\mathbf{u})=P(\mathbf{u})/\Phi_{S}(\mathbf{u})$, where
$\Phi_{S}(\mathbf{u})=\prod_{j=1}^{N}l_{j}(\mathbf{u})$, we want to solve the
inverse moment problem. (It is easy to obtain $P(\mathbf{u})$ from information
on the moments of order at most $N-d-1$ using Lemma 11.)

To solve the latter inverse problem using Corollary 3, we need to find an
appropriate set of weights
$\mathbf{w}=\\{w_{i_{1},\ldots,i_{d}}\\},\;i_{1}<i_{1}<\ldots<i_{d}$, where
$w_{i_{1},\ldots,i_{d}}$ is the weight (recall Definition 1.14) of the
$d$-dimensional simplex
$\mathrm{conv}(\mathbf{v}_{i_{1}},\mathbf{v}_{i_{2}},\ldots,\mathbf{v}_{i_{d}},\mathbf{v}_{N})$
so that

$\sum_{i_{1}<i_{2}<\ldots<i_{d}}\frac{w_{i_{1},\ldots,i_{d}}}{l_{i_{1}}(\mathbf{u})l_{i_{2}}(\mathbf{u})\dots
l_{i_{d}}(\mathbf{u})l_{{}_{N}}(\mathbf{u})}=\frac{P(\mathbf{u})}{l_{1}(\mathbf{u})l_{2}(\mathbf{u})\dots
l_{{}_{N}}(\mathbf{u})}.$

Clearing the denominators, we get the equation

$\sum_{i_{1}<i_{2}<\ldots<i_{d}}{w_{i_{1},\ldots,i_{d}}}{l_{j_{1}}(\mathbf{u})l_{j_{2}}(\mathbf{u})\dots
l_{j_{N-d-1}}(\mathbf{u})l_{N}(\mathbf{u})}={P(\mathbf{u})},$

where $\\{j_{1},\ldots
j_{N-d-1}\\}=\\{1,2,\ldots,N-1\\}\setminus\\{i_{1},\ldots,i_{d}\\}$. The
latter equation is obviously equivalent to the system of linear equations

$Mat_{S}\cdot\mathbf{w}=(p_{I_{1}},\dots,p_{I_{t}}),\quad\text{where\
}P(\mathbf{u})=\sum_{I}p_{I}\mathbf{u}^{I},$

and $\mathbf{w}=\\{w_{i_{1},\ldots,i_{d}}\\}$ is the vector consisting of the
weights of all simplices containing $\mathbf{v}_{N}$. ∎

Theorem 12 solves the inverse moment problem for strongly non-degenerate
spanning set $S$. We can make this solution more explicit by giving a closed
formula for the inverse matrix $Mat_{S}^{-1}$. To do this we introduce an
extra variable $u_{0}\in\mathbb{R}$ and identify the space $Pol(N-d-1,d)$ with
the space $HPol(N-d-1,d+1)$ of homogeneous forms of degree $N-d-1$ in $d+1$
variables $(u_{0},u_{1},\dots,u_{d})$. We homogenize each linear form
$l_{i}(\mathbf{u})$ in $\mathfrak{L}$ as
$l_{i}(\mathbf{u},u_{0})=u_{0}-\langle\mathbf{v}_{i},\mathbf{u}\rangle$. (The
matrix $Mat_{S}$ remains unchanged.)

We also need the following $(d+1)\times(N-1)$ matrix $\mathbb{L}$

$\mathbb{L}=\bordermatrix{~{}&l_{1}&l_{2}&\dots&l_{N-1}\cr
u_{0}&1&1&\dots&1\cr
u_{1}&-\langle\mathbf{v}_{1},e_{1}\rangle&-\langle\mathbf{v}_{2},e_{1}\rangle&\dots&-\langle\mathbf{v}_{N-1},e_{1}\rangle\cr~{}\vdots&\vdots&\vdots&&\vdots\cr
u_{d}&-\langle\mathbf{v}_{1},e_{d}\rangle&-\langle\mathbf{v}_{2},e_{d}\rangle&\dots&-\langle\mathbf{v}_{N-1},e_{d}\rangle\cr}$
(3.1)

associated with $Mat_{S}$. For all possible subsets
$\mathbf{i}[d]=\\{i_{1},\dots,i_{d}\\},$ $i_{1}<i_{2}<\dots<i_{d}$ of $d$
distinct columns of $\mathbb{L}$ consider the linear in $\mathbf{u}$ function
$\mathbb{L}_{\mathbf{i}[d]}(\mathbf{u})$ given by:

$\mathbb{L}_{\mathbf{i}[d]}(\mathbf{u})=\text{det}\begin{bmatrix}u_{0}&1&\dots&1\\\
u_{1}&-\langle\mathbf{v}_{i_{1}},e_{1}\rangle&\dots&-\langle\mathbf{v}_{i_{d}},e_{1}\rangle\\\
\vdots&&\dots&\\\
u_{d}&-\langle\mathbf{v}_{i_{1}},e_{d}\rangle&\dots&-\langle\mathbf{v}_{i_{d}},e_{d}\rangle\end{bmatrix}.$
(3.2)

Denote by $\mathbb{L}_{\mathbf{i}[d],j}$ the coefficient of $u_{j}$ in the
linear form $\mathbb{L}_{\mathbf{i}[d]}(\mathbf{u})$. For each $1\leq j\leq
N-1$ define $\mathbb{L}(j,\mathbf{i}[d])$ as

$\mathbb{L}(j,\mathbf{i}[d])=\text{det}\begin{bmatrix}1&1&\dots&1\\\
-\langle\mathbf{v}_{j},e_{1}\rangle&-\langle\mathbf{v}_{i_{1}},e_{1}\rangle&\dots&-\langle\mathbf{v}_{i_{d}},e_{1}\rangle\\\
\vdots&&\dots&\\\
-\langle\mathbf{v}_{j},e_{d}\rangle&-\langle\mathbf{v}_{i_{1}},e_{d}\rangle&\dots&-\langle\mathbf{v}_{i_{d}},e_{d}\rangle\end{bmatrix}.$
(3.3)

Note that if $j\notin\mathbf{i}[d]$ then $\mathbb{L}(j,\mathbf{i}[d])\neq 0$,
as by the assumption of strong non-degeneracy of $S$ the corresponding $d+1$
linear forms are linearly independent. On the other hand, if
$j\in\mathbf{i}[d]$ then we have $\mathbb{L}(j,\mathbf{i}[d])=0.$

The matrix $Mat_{S}^{-1}$ has the following explicit description.

######  Lemma 18.

For each $(N-d-1)$-tuple of forms
$\\{l_{j_{1}},l_{j_{2}},\dots,l_{j_{N-d-1}}\\}$ set
$\mathbf{i}[d]=\\{i_{1},\dots,i_{d}\\}=[N-1]\setminus\\{j_{1},\dots,j_{N-d-1}\\},$
where $i_{1}<i_{2}\dots<i_{d}.$ Then,

$Mat_{S}^{-1}=\bordermatrix{~{}&\dots&u_{0}^{n_{0}}u_{1}^{n_{1}}\dots
u_{d}^{n_{d}}&\dots\cr\quad\vdots&&\quad\vdots&\cr l_{j_{1}}\dots
l_{j_{N-d-1}}&\dots&\frac{\prod\limits_{j=0}^{d}\mathbb{L}_{\mathbf{i}[d],j}^{n_{j}}}{\prod\limits_{k=1}^{N-d-1}\mathbb{L}(j_{k},\mathbf{i}[d])}&\dots\cr\quad\vdots&&\quad\vdots&\cr}.$
(3.4)

###### Proof of Lemma 18.

In order to show that $Mat_{S}^{-1}$ defined by (3.4) is indeed the inverse of
$Mat_{S}$ we need to verify that $Mat_{S}^{-1}\cdot Mat_{S}$ is the identity
operator on $V_{\mathfrak{L}}$.

Let $\mathbf{e}^{\prime}$ be the standard basis vector of $V_{\mathfrak{L}}$
corresponding to the product of linear forms $l_{j^{\prime}_{1}}\dots
l_{j^{\prime}_{N-d-1}}$. Then $Mat_{S}\cdot\mathbf{e}^{\prime}$ is the vector
consisting of the monomial coefficients of the homogeneous form
$l_{j^{\prime}_{1}}\dots l_{j^{\prime}_{N-d-1}}$ in the variables
$u_{0},\dots,u_{d}$. Let $\mathbf{e}^{T}$ be the row vector of $Mat_{S}^{-1}$
corresponding to the product $l_{j_{1}}\dots l_{j_{N-d-1}}$. We note that in
$\mathbf{e}^{T}\cdot\left(Mat_{S}\cdot\mathbf{e}^{\prime}\right)$ one can
factor out the common denominator
$\prod\limits_{k=1}^{N-d-1}\mathbb{L}(j_{k},\mathbf{i}[d])$ of all fractions
in $\mathbf{e}^{T}$; the remaining factor is of the form

$\sum\limits_{\begin{subarray}{c}I=(n_{0},\dots,n_{d})\\\
|I|=N-d-1\end{subarray}}Mat_{S}[\mathbf{u}^{I},\mathbf{e}^{\prime}]\prod\limits_{j=0}^{d}\mathbb{L}_{\mathbf{i}[d],j}^{n_{j}}=l_{j^{\prime}_{1}}(\mathbb{L}_{\mathbf{i}[d]}(\mathbf{u}))\dots
l_{j^{\prime}_{N-d-1}}(\mathbb{L}_{\mathbf{i}[d]}(\mathbf{u})).$ (3.5)

Note that
$l_{j^{\prime}_{k}}(\mathbb{L}_{\mathbf{i}[d]}(\mathbf{u}))=\mathbb{L}(j^{\prime}_{k},\mathbf{i}[d])$
for $1\leq k\leq N-d-1$, i.e. this holds for all terms in the product on the
right-hand side of (3.5). Hence, if $\mathbf{e}\neq\mathbf{e}^{\prime}$ then
$\mathbf{e}^{T}\cdot\left(Mat_{S}\cdot\mathbf{e}^{\prime}\right)=0$, as among
$j^{\prime}_{1},\dots,j^{\prime}_{N-d-1}$ one can find
$j^{\prime}\in\mathbf{i}[d]$ with $\mathbb{L}(j^{\prime},\mathbf{i}[d])=0$. On
the other hand, if $\mathbf{e}$ and $\mathbf{e}^{\prime}$ coincide, then the
right-hand side of (3.5) is equal to
$\prod\limits_{k=1}^{N-d-1}\mathbb{L}(j_{k},\mathbf{i}[d])$. Dividing by the
common denominator of the fractions in $\mathbf{e}$, we obtain
$\mathbf{e}^{T}\cdot\left(Mat_{S}\cdot\mathbf{e}\right)=1.$ ∎

##  4\. Inverse moment problem for weakly non-degenerate $S$

Given an arbitrary spanning set
$S=\\{\mathbf{v}_{1},\mathbf{v}_{2},\ldots,\mathbf{v}_{N}\\}$, consider the
linear space $\Theta(S)\subseteq Pol(N-d-1,d)$ spanned by all products
$l_{j_{1}}l_{j_{2}}\ldots l_{j_{N-d-1}},\;j_{1}<j_{2}<\ldots<j_{N-d-1}$. 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 $S$.

######  Lemma 19.

$\Theta(S)=Pol(N-d-1,d)$ if and only if $S$ is weakly non-degenerate, i.e.
each $(d+2)$-tuple of points of $S$ is spanning.

###### Proof.

We have $N$ (non-homogeneous) linear forms $l_{1}\dots,l_{N}$ in variables
$\mathbf{u}=(u_{1},\dots,u_{d})$ and the linear space $V_{\mathfrak{L}}$
spanned by all possible products of $(N-d-1)$-tuples of distinct forms. We
need to investigate whether $V_{\mathfrak{L}}$ coincides with $Pol(N-d-1,d)$.
Homogenizing, we consider the same question for the linear homogeneous forms
and the homogeneous polynomials of degree $N-d-1$ in variables
$(u_{0},u_{1},\dots,u_{d})$.

First assume that there are $d+2$ linear forms $l_{1},\dots,l_{d+2}$ which are
not spanning. Then one can find a non-zero vector
$\mathbf{z}_{0}\in\mathbb{R}^{d+1}$, such that $l_{1}(z)=\dots=l_{d+2}(z)=0$.
Note that each product of $N-d-1$ different forms chosen from
$l_{1},\dots,l_{N}$ contains at least one form among
$\\{l_{1},\dots,l_{d+2}\\}$. Therefore any linear combination of products of
$N-d-1$ forms vanishes at $\mathbf{z}_{0}$. Thus $V_{\mathfrak{L}}$ cannot
coincide with $HPol(N-d-1,d+1)$.

Conversely, assume that every $(d+2)$-tuple of distinct forms among
$l_{1},\dots,l_{N}$ is spanning. First, we notice that $HPol(N-d-1,d+1)$ 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 $\mathbb{R}^{d+1}$, 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 $l_{1}^{i_{1}}\dots l_{N}^{i_{N}}$ 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
$\mathbf{m}=l_{1}^{k_{1}}\dots l_{N}^{k_{N}}$ having the maximal number of
distinct forms in the product. Since $\mathbf{m}$ is not a product of all
distinct forms, it should contain a form $l_{i}$ in some power $k_{i}\geq 2$.
Given that $k_{i}\geq 2$ and the degree of $\mathbf{m}$ is $N-d-1$, one can
find $d+2$ distinct forms $l_{i_{1}},\dots,l_{i_{d+2}}$ that do not appear in
$\mathbf{m}$. Since any $d+2$ of our forms span the dual space of
$\mathbb{R}^{d+1}$, we can express $l_{i}$ as a linear combination of
$l_{i_{1}},\dots,l_{i_{d+2}}$. Now rewrite $\mathbf{m}$ as
$\left(\alpha_{1}\cdot l_{i_{1}}+\dots+\alpha_{d+2}\cdot
l_{i_{d+2}}\right)l_{1}^{k_{1}}\dots l_{i}^{k_{i}-1}\dots l_{N}^{k_{N}}$,
where $\alpha_{1}\cdot l_{i_{1}}+\dots+\alpha_{d+2}\cdot l_{i_{d+2}}=l_{i}.$
Thus we get an expression of $\mathbf{m}$ as a linear combination of monomials
$\alpha_{j}\cdot l_{i_{j}}l_{1}^{k_{1}}\dots l_{i}^{k_{i}-1}\dots
l_{N}^{k_{N}}$, where each such monomial has more distinct forms than
$\mathbf{m}$. Each of such monomials can be expressed as a linear combination
of products of all distinct forms, since $\mathbf{m}$ 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 $\mathbf{m}$ 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 $\mathbf{v}_{N}$. Because of that we need to
consider all $N$ points and include one more linear form $l_{N}$ into the
corresponding matrix $\mathbb{L}$. Slightly abusing our notation, we denote by
$\mathbb{L}$ the same matrix as before, although it contains one more (last)
column corresponding to $\mathbf{v}_{N}$. Similarly to notation (3.3), for a
given set $J$ of $d+1$ linear forms we denote by $\mathbb{L}(J)$ the
determinant of the corresponding $(d+1)\times(d+1)$-minor of $\mathbb{L}$.

We introduce the extended $\binom{N-1}{d}\times\binom{N}{d+1}$-matrix
$\widetilde{Mat}_{S}$ with columns consisting of the coefficients of the
homogeneous polynomial $l_{i_{1}}(\mathbf{u})\dots l_{i_{N-d-1}}(\mathbf{u})$
with respect to the monomial basis in the variables
$(u_{0},u_{1},\dots,u_{d}).$ By Lemma 19, $\widetilde{Mat}_{S}$ has full rank,
since it determines a surjective linear map onto $HPol(N-d-1,d+1)$. Thus
$\widetilde{Mat}_{S}$ has a maximal minor with a non-vanishing determinant.
Formula (4.1) holds for the determinant of any maximal minor of
$\widetilde{Mat}_{S}$.

######  Lemma 20.

Let $\mathfrak{S}$ be any set of $\binom{N-1}{d}$ columns of
$\widetilde{Mat}_{S}$. We label each column $T\in\mathfrak{S}$ by the
corresponding subset of the linear forms $l_{1},\dots,l_{N}$ of cardinality
$N-d-1$. Then the determinant of the maximal minor
$\widetilde{Mat}_{S}(\mathfrak{S})$ formed by the columns of $\mathfrak{S}$ is
given by:

$\det\left[\widetilde{Mat}_{S}(\mathfrak{S})\right]=k(\mathfrak{S})\cdot\prod_{\begin{subarray}{c}J\in{N\brack
d+1}:\\\ \forall T\in\mathfrak{S}~{}T\cap
J\neq\emptyset\end{subarray}}\mathbb{L}(J),$ (4.1)

where $k(\mathfrak{S})$ is a constant (possibly equal to zero) depending only
on the combinatorial structure of the $(N-d-1)$-tuples in the set
$\mathfrak{S}$.

###### Proof.

Fix the set $\mathfrak{S}$ as above. In what follows we treat both sides of
(4.1) as complex-valued polynomials in $N\cdot(d+1)$ variables, these
variables being the entries of matrix $\mathbb{L}$.

We first show that every determinant $\mathbb{L}(J)$ divides
$\det\left[\widetilde{Mat}_{S}(\mathfrak{S})\right]$. Indeed, let
$J=\\{j_{1},\dots,j_{d+1}\\}$ be a set of $(d+1)$ forms which has a nonempty
intersection with any $(N-d-1)$-tuple of forms in $\mathfrak{S}$. Let
$\mathbf{z}=(z_{1},\dots,z_{N\cdot(d+1)})$ be a zero of the polynomial
$\mathbb{L}(J)$, which means that forms $l_{j_{1}},\dots,l_{j_{d+1}}$
comprised of the corresponding coordinates of $\mathbf{z}$ are linearly
dependent. Therefore, there is a non-zero vector
$\mathbf{u}_{0}\in\mathbb{R}^{d+1}$, such that
$l_{j_{1}}(\mathbf{u}_{0})=\dots=l_{j_{d+1}}(\mathbf{u}_{0})=0$. Consider the
row vector $(\mathbf{u}_{0}^{I})$ consisting of $\binom{N-1}{d}$ homogeneous
monomials of degree $N-d-1$ evaluated at $\mathbf{u}_{0}.$ We notice that
$(\mathbf{u}_{0}^{I})$ is in the kernel of
$\widetilde{Mat}_{S}(\mathfrak{S})$, as the product of $(\mathbf{u}_{0}^{I})$
with each column vector $T\in\mathfrak{S}$ of
$\widetilde{Mat}_{S}(\mathfrak{S})$ is equal to $\prod_{j\in
T}l_{j}(\mathbf{u}_{0})$; and every set $T\in\mathfrak{S}$ contains at least
one of the forms $l_{j_{1}},\dots,l_{j_{d+1}}$ in such a product. Thus
$\det\left[\widetilde{Mat}_{S}(\mathfrak{S})\right]$ also vanishes at such
$\mathbf{z}$.

We recall a well-known fact (cf. e.g. [Boch]*Theorem 61.1) that

$\mathbb{L}(J)=\det\bordermatrix{~{}&l_{j_{1}}&l_{j_{2}}&\dots&l_{j_{d+1}}\cr
u_{0}&z_{j_{1}}&z_{j_{2}}&\cdots&z_{j_{d+1}}\cr
u_{1}&z_{j_{1}+N}&z_{j_{2}+N}&\cdots&z_{j_{d+1}+N}\cr\vdots&\vdots&\vdots&\ddots&\vdots\cr
u_{d}&z_{j_{1}+dN}&z_{j_{2}+dN}&\cdots&z_{j_{d+1}+dN}\cr},$

is an irreducible complex-valued polynomial in variables
$z_{j_{1}},z_{j_{2}}\dots,z_{j_{d+1}+Nd}$. Now if every zero of an irreducible
polynomial $p(z_{1},\dots,z_{N(d+1)})$ annihilates another polynomial
$q(z_{1},\dots,z_{N(d+1)})$ then $p$ divides $q$. We conclude that
$\mathbb{L}(J)$ divides $\det\left[\widetilde{Mat}_{S}(\mathfrak{S})\right]$.

Using the fact that each $\mathbb{L}(J)$ is an irreducible polynomial and all
$\mathbb{L}(J)$’s are pairwise distinct (i.e. have distinct sets of projective
zeros) we conclude that the product of $\mathbb{L}(J)$’s in the right-hand
side of (4.1) divides $\det\left[\widetilde{Mat}_{S}(\mathfrak{S})\right].$

Finally, the product of $\mathbb{L}(J)$’s has the degree

$(d+1)\left|\left.\left\\{J\in{N\brack d+1}\right|\forall
T\in\mathfrak{S}\quad T\cap J\neq\emptyset\right\\}\right|.$ (4.2)

We observe that for each $T\in\mathfrak{S}$ 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 $\binom{N}{d+1}$ $(d+1)$-tuples except those
$\binom{N-1}{d}$ complements of a $T\in\mathfrak{S}$. Therefore, (4.2) equals

$(d+1)\left(\binom{N}{d+1}-\binom{N-1}{d}\right)=(d+1)\binom{N-1}{d+1}=(N-d-1)\binom{N-1}{d}.$

The latter expression coincides with the degree of the polynomial
$\det\left[\widetilde{Mat}_{S}(\mathfrak{S})\right]$ (assuming that it is not
a zero), as $\widetilde{Mat}_{S}(\mathfrak{S})$ has $\binom{N-1}{d}$ columns
and each entry is a homogeneous polynomial of degree $N-d-1$.

Hence $\widetilde{Mat}_{S}(\mathfrak{S})$ coincides with the product of
$\mathbb{L}(J)$’s up to a constant factor which might vanish. This constant
does not depend on the entries of matrix $\mathbb{L}$ and hence it is
completely determined by the set $\mathfrak{S}$, regardless of the location of
points of $S$ in $\mathbb{R}^{d}$. ∎

Lemma 19 allows us to solve the inverse moment problem for a given weakly non-
degenerate $S=\\{\mathbf{v}_{1},\dots,\mathbf{v}_{N}\\}$ in a certain linear
space $\widetilde{\mathfrak{M}}(S)\supseteq\mathfrak{M}^{\Delta}(S)$ of
measures supported on $\mathrm{conv}(S)$. Namely,
$\widetilde{\mathfrak{M}}(S)$ is spanned by measures
$\mu\in\widetilde{\mathfrak{M}}(S)$ whose normalized moment generating
functions $F_{\mu}(\mathbf{u})$ belong to $\mathfrak{Rat}(S)$, i.e.
$F_{\mu}(\mathbf{u})=P(\mathbf{u})/\Phi_{S}(\mathbf{u})$, where
$\Phi_{S}(\mathbf{u})=\prod_{j=1}^{N}l_{j}(\mathbf{u})$ and $P(\mathbf{u})$ is
a polynomial of degree at most $N-d-1$. Indeed, by Lemma 19 any
$R(\mathbf{u})\in\mathfrak{Rat}(S)$ can be represented in the form

$R(\mathbf{u})=\sum_{i_{1}<i_{2}<\dots<i_{d+1}\leq N}\frac{K_{i_{1}i_{2}\dots
i_{d+1}}}{l_{i_{1}}l_{i_{2}}\dots l_{i_{d+1}}},$ (4.3)

with some real constants $K_{i_{1}i_{2}\dots i_{d+1}}$. If
$\mathbf{v}_{i_{1}},\mathbf{v}_{i_{2}},\dots,\mathbf{v}_{i_{d+1}}$ span
$\mathbb{R}^{d}$ then the term $\frac{K_{i_{1}i_{2}\dots
i_{d+1}}}{l_{i_{1}}l_{i_{2}}\dots l_{i_{d+1}}}$ 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 $\mathbf{v}_{i_{1}},\mathbf{v}_{i_{2}},\dots,\mathbf{v}_{i_{d+1}}$ only
span a hyperplane $H$ in $\mathbb{R}^{d}$ then (4.3) corresponds to a singular
(w.r.t. to the Lebesgue measure on $\mathbb{R}^{d}$) measure $\mu_{\delta}$
supported on
$\delta=\mathrm{conv}(\mathbf{v}_{i_{1}},\dots,\mathbf{v}_{i_{d+1}})$. One way
to define it as the weak limit of a sequence of (absolutely continuous with
respect to the Lebesgue measure on $\mathbb{R}^{d}$) measures—the
appropriately scaled standard measures $\mu_{\delta_{t}}$ of family of
$d$-dimensional simlices $\delta_{t}$ which degenerate into $\delta$ when
$t=0$. There is no loss in generality in assuming $K_{i_{1}i_{2}\dots
i_{d+1}}=1$, i.e. to deal with probability measures.

######  Proposition 21.

Let $\mathbf{W}=\\{\mathbf{w}_{1},\dots,\mathbf{w}_{d},\mathbf{w}_{d+1}\\}$ be
a $(d+1)$-tuple of points in $\mathbb{R}^{d}$ such that $\mathbf{W}$ spans a
hyperplane $H\subset\mathbb{R}^{d}$. Denote by
$l_{\mathbf{w}_{1}}=1-\langle\mathbf{w}_{1},\mathbf{u}\rangle,\dots,l_{\mathbf{w}_{d+1}}=1-\langle\mathbf{w}_{d+1},\mathbf{u}\rangle$
the associated linear forms. There exists a unique measure $\mu_{\mathbf{W}}$
supported on $\delta=\mathrm{conv}({\mathbf{W}})$ with the normalized moment
generating function $F_{\mu_{\mathbf{W}}}(\mathbf{u})$ given by

$F_{\mu_{\mathbf{W}}}(\mathbf{u})=\frac{1}{l_{\mathbf{w}_{1}}l_{\mathbf{w}_{2}}\dots
l_{\mathbf{w}_{d+1}}}.$ (4.4)

###### Proof.

Without loss of generality assume that
$\mathbf{W}=\\{\mathbf{w}_{1},\dots,\mathbf{w}_{d},\mathbf{w}_{d+1}\\}$ is
ordered in such a way that $\\{\mathbf{w}_{1},\dots,\mathbf{w}_{d}\\}$ span
$H$. Then, $\delta_{t}$ is defined as
$\delta_{t}=\mathrm{conv}(\delta,\mathbf{w}_{i_{d+1}}+t\mathbf{z})$, with
$\mathbf{z}$ a unit normal to $H$, and $\mu_{\delta_{t}}$ as the uniform
density probability measure supported on $\delta_{t}$. Then $\lim\limits_{t\to
0}\mu_{\delta_{t}}=\mu_{\delta}$, where $\lim$ is understood in sense of _weak
convergence of distributions (measures)_ , i.e. that $\lim\limits_{t\to 0}\int
fd\mu_{\delta_{t}}=\int fd\mu_{\delta}$ for any bounded, continuous real
function on $\mathbb{R}^{d}$, cf. e.g. [MR1700749]. Then, this measure has
compact support, and thus is determined by its moments, cf. e.g.
[MR2244695]*Proposition 3.2. ∎

######  Remark 12.

One can prove that the integration of a smooth compactly supported function
$\phi$ with respect to the limiting measure $\mu_{\mathbf{W}}$ is given by the
integration of $\phi$ over $\delta$ with a continuous piecewise linear weight
function uniquely determined by $\delta$. Similar limits appear frequently in
the theory of splines. Since we only need the existence of $\mu_{\mathbf{W}}$
we do not pursue this topic here.

Our solution of the inverse moment problem for the linear space
$\widetilde{\mathfrak{M}}(S)$ closely follows the pattern presented in Example
2. In other words, given a weakly non-degenerate $S$ and the set of moments up
to order $N-d-1$ we

  1. (i)

produce the rational function $R(\mathbf{u})\in\mathfrak{Rat}(S)$ with Taylor
coefficients coinciding with the normalized moments;

  2. (ii)

represent $R(\mathbf{u})$ in the form (4.3);

  3. (iii)

for each term as in (4.4) determine the underlying measure supported on the
(probably degenerate) convex hull of the vertices
$\mathbf{v}_{i_{1}},\mathbf{v}_{i_{2}},\dots,\mathbf{v}_{i_{d+1}}$.

We can now prove our central result claiming that
$\mathfrak{M}^{\Delta}(S)=\mathfrak{M}(S)$ 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
$F_{\mathcal{P}}(\mathbf{u})$ for an arbitrary generalized polytope
$\mathcal{P}$ with the vertex set $S$ is of the form $\Pi_{i=1}^{N}l_{i}$ by
Proposition 6, and its numerator belongs to $Pol(N-d-1,d)$. As $S$ is weakly
non-degenerate, $F_{\mathcal{P}}(\mathbf{u})$ can be written as a linear
combination of the fractions as in (4.3), where $(i_{1},i_{2},\ldots i_{d+1})$
runs over the set of $(d+1)$-tuples of indices. If a $(d+1)$-tuple
$l_{i_{1}},l_{i_{2}},\ldots l_{i_{d+1}}$ is spanning then
$\frac{K}{l_{i_{1}}l_{i_{2}}\ldots l_{i_{d+1}}}$ is the moment generating
function of the measure supported on the simplex $\Delta$, determined by its
denominator, with the uniform density $K/d!\text{Vol}(\Delta)$. By Proposition
21, if a $(d+1)$-tuple $l_{i_{1}},l_{i_{2}},\ldots l_{i_{d+1}}$ is not
spanning then $\frac{K}{l_{i_{1}}l_{i_{2}}\ldots l_{i_{d+1}}}$ is the moment
generating function of a singular measure supported on a degenerate simplex.
As $\mathcal{P}$ 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 $\mu$ of a polytope with the vertex set $S$ then we never
obtain degenerate simplices while solving the inverse moment problem. This is
why $\mathfrak{M}^{\Delta}(S)=\mathfrak{M}(S)$. However, an explicit
description of $\mathfrak{F}^{\Delta}(S)$ 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 $\dim\mathfrak{M}^{\Delta}(S)$ and describes a
procedure to construct a basis for $\mathfrak{M}^{\Delta}(S)$.

######  Proposition 22.

Let $S=\\{\mathbf{v}_{1},\ldots,\mathbf{v}_{N}\\}\subset\mathbb{R}^{d}$ be an
arbitrary weakly non-degenerate spanning set. Then

  1. (i)

$\dim\mathfrak{M}^{\Delta}(S)=\binom{N-1}{d}-\sharp_{deg}$ where
$\sharp_{deg}$ is the number of degenerate simplices, i.e. the number of non-
spanning $(d+1)$-tuples of points of $S$.

  2. (ii)

If $\delta$ is a degenerate $d$-dimensional simplex with vertices in
$S\setminus\\{\mathbf{v}_{i}\\}$ then there is exactly one linear dependence
among the standard measures of all $d$-dimensional simplices on
$\mathbf{v}_{i}$ and $d$ vertices of $\delta$.

  3. (iii)

The standard measure of any $d$-dimensional simplex on $\mathbf{v}_{i}$ is
contained in at most one dependence as in (ii).

  4. (iv)

For any vertex $\mathbf{v}_{i}$ one can construct a (in general, non-unique)
basis $\mathcal{B}_{i}$ of $\mathfrak{M}^{\Delta}(S)$ consisting of standard
measures of $d$-dimensional simplices on $\mathbf{v}_{i}$, as follows.

    1. (a)

Start from the set $\mathcal{B}_{i}$ of the $d$-dimensional simplices on
$\mathbf{v}_{i}$.

    2. (b)

For each degenerate simplex $\delta$ not containing $\mathbf{v}_{i}$, remove
from $\mathcal{B}_{i}$ the standard measure of an arbitrary simplex on
$\mathbf{v}_{i}$ from the corresponding to $\delta$ linear dependence, cf.
(i).

Thus we obtain $\binom{N-1}{d}-\sharp_{deg}$ standard measures of
$d$-dimensional simplices, forming a basis of $\mathfrak{M}^{\Delta}(S)$.

###### Proof.

To prove (i), notice that
$\dim\widetilde{\mathfrak{M}}(S)=\dim\mathfrak{Rat}(S)=\binom{N-1}{d}$. As
well,
$\widetilde{\mathfrak{M}}(S)=\mathfrak{M}(S)\oplus\mathfrak{M}_{deg}(S)$,
where $\mathfrak{M}_{deg}(S)$ is the linear span of the measures
$\mu_{\delta}^{(1)}$ with $\delta$ running over the set of all degenerate
simplices spanned by $(d+1)$-tuples of dependent vertices in $S$, cf.
Proposition 21. Observe that these measures $\mu_{\delta}^{(1)}$ 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 $\Sigma_{0}$ be a dependent $d+1$-subset of $S$, and
$\delta=\mathrm{conv}(\Sigma_{0})$ be as in (ii). Then $\delta$ spans a
hyperplane $H_{0}$. As each $d$-dimensional simplex on
$\sigma_{0}:=\mathbf{v}_{i}$ and $d$ vertices from $\Sigma_{0}$ is uniquely
defined by the latter, it suffices to analyze dependencies between the
standard measures of $d-1$-simplices with vertices in $\Sigma_{0}$.

We can view $\Sigma_{0}$ as a weakly non-degenerate subset in
$\mathbb{R}^{d-1}\cong H_{0}$. By (i), we have
$\dim\mathfrak{M}^{\Delta}(\Sigma_{0})=\binom{d}{d-1}-\sharp_{deg}(\Sigma_{0})$.
If $\Sigma_{0}$ is strongly non-degenerate as a subset of
$H_{0}\cong\mathbb{R}^{d-1}$, i.e. $\sharp_{deg}(\Sigma_{0})=0$, then
$\dim\mathfrak{M}^{\Delta}(\Sigma_{0})=d$, i.e. there is exactly one linear
dependence between the standard measures of $d-1$-simplices with vertices in
$\Sigma_{0}$, and we are done. Otherwise,
$\Sigma_{0}=\\{\sigma_{1}\\}\cup\Sigma_{1}$, with $\Sigma_{1}$ spanning a
hyperplane $H_{1}$ in $H_{0}$. Moreover, this can only happen if $d\geq 3$.
Now, we can repeat the whole argument with $\sigma_{1}$ in place of
$\sigma_{0}$, $\Sigma_{1}$ in place of $\Sigma_{0}$, and $H_{1}$ in place of
$H_{0}$. Again, we either have $\Sigma_{1}$ strongly degenerate, and we are
done, or we repeat this argument, etc., until we hit a strongly non-degenerate
$\Sigma_{k}$, 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 $\mathbf{v}_{i}$ distinct from $\mathbf{v}_{i}$ span a hyperplane, and the
only possibility for a degenerate simplex $\delta$ as in (ii) is to lie in
this hyperplane.

Finally, to prove (iv), observe that the set $\mathcal{B}_{i}^{\prime}$ of the
standard measures of $d$-dimensional simplices containing a given vertex
$\mathbf{v}_{i}$ always spans $\mathfrak{M}^{\Delta}(S)$, see Lemma 15. Now
for each degenerate $d$-simplex $\delta$ we prune $\mathcal{B}_{i}^{\prime}$
by removing the standard measure of a simplex in the linear dependence
corresponding to $\delta$. In view of (ii) and (iii) this process is well-
defined and unambiguous. In the end we obtain $\binom{N-1}{d}-\sharp_{deg}$
standard measures of $d$-dimensional simplices. In view of (i) they form a
basis of $\mathfrak{M}^{\Delta}(S)$, as claimed. ∎

######  Remark 14.

The above discussions show that the columns of $\widetilde{Mat}_{S}$
corresponding to degenerate simplices must necessarily be included in any non-
vanishing maximal minor $\widetilde{Mat}_{S}(\mathfrak{S})$.

We conclude our discussion of the weakly non-degenerate case with two
examples.

$\mathbf{v}_{5}=(0,0)$$\mathbf{v}_{2}=(2,0)$$\mathbf{v}_{1}=(1,1)$$\mathbf{v}_{3}=(2,2)$$\mathbf{v}_{4}=(0,2)$$\mathbf{v}_{1}=\mathbf{v}_{5}=(0,0)$$\mathbf{v}_{2}=(2,0)$$\mathbf{v}_{3}=(1,1)$$\mathbf{v}_{4}=(0,2)$
Figure 3. Vertices for Examples 3 and 4.

######  Example 3.

Let
$S=\\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3},\mathbf{v}_{4},\mathbf{v}_{5}\\}$
where
$\mathbf{v}_{1}=(1,1),\mathbf{v}_{2}=(2,0),\mathbf{v}_{3}=(2,2),\mathbf{v}_{4}=(0,2),\mathbf{v}_{5}=(0,0)$.
Then
$l_{1}=1-u_{1}-u_{2},l_{2}=1-2u_{1},l_{3}=1-2u_{1}-2u_{2},l_{4}=1-2u_{2},l_{5}=1$.
Calculating the products $l_{i}l_{j},\,i<j$ and taking their coefficients in
the standard monomial basis of $Pol(2,2)$ we obtain the following $6\times
10$-matrix $\widetilde{Mat}_{S}$

$\widetilde{Mat}_{S}=\bordermatrix{~{}&l_{1}l_{2}&l_{1}l_{3}&l_{1}l_{4}&l_{1}l_{5}&l_{2}l_{3}&l_{2}l_{4}&l_{2}l_{5}&l_{3}l_{4}&l_{3}l_{5}&l_{4}l_{5}\cr
1&1&1&1&1&1&1&1&1&1&1\cr u_{1}&-3&-3&-1&-1&-4&-2&-2&-2&-2&0&\cr
u_{2}&-1&-3&-3&-1&-2&-2&0&-4&-2&-2\cr u_{1}^{2}&2&2&0&0&4&0&0&0&0&0\cr
u_{1}u_{2}&2&4&2&0&4&4&0&4&0&0\cr u_{2}^{2}&0&2&2&0&0&0&0&4&0&0}.$

Its rank equals $6$ and one of non-vanishing maximal minors consists of the
columns with numbers $\mathfrak{S}=\\{5,6,7,8,9,10\\}$. (Recall that any non-
vanishing maximal minor must include columns 6 and 9 corresponding to
degenerate triples $(\mathbf{v}_{1},\mathbf{v}_{3},\mathbf{v}_{5})$ and
$(\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{4})$ resp.) The corresponding
submatrix $\widetilde{Mat}_{S}(\mathfrak{S})$ equals

$\widetilde{Mat}_{S}(\mathfrak{S})=\bordermatrix{~{}&l_{2}l_{3}&l_{2}l_{4}&l_{2}l_{5}&l_{3}l_{4}&l_{3}l_{5}&l_{4}l_{5}\cr
1&1&1&1&1&1&1\cr u_{1}&-4&-2&-2&-2&-2&0\cr u_{2}&-2&-2&0&-4&-2&-2&\cr
u_{1}^{2}&4&0&0&0&0&0\cr u_{1}u_{2}&4&4&0&4&0&0\cr u_{2}^{2}&0&0&0&4&0&0}.$

Further,

$4\widetilde{Mat}_{S}^{-1}(\mathfrak{S})=\bordermatrix{~{}&1&u_{1}&u_{2}&u_{1}^{2}&u_{1}u_{2}&u_{2}^{2}\cr
l_{2}l_{3}&0&0&4&0&-4&4\cr l_{2}l_{4}&0&0&0&0&-2&2\cr
l_{2}l_{5}&0&0&2&0&-2&0&\cr l_{3}l_{4}&1&-1&0&0&-1&1\cr
l_{3}l_{5}&0&1&0&0&-1&0&\cr l_{4}l_{5}&0&-1&1&1&-1&0}.$

Thus, given an arbitrary rational function
$R(u_{1},u_{2})=P(u_{1},u_{2})/\Phi_{S}(u_{1},u_{2})$ where
$P(u_{1},u_{2})=a_{00}+a_{1,0}u_{1}+a_{0,1}u_{2}+a_{2,0}u_{1}^{2}+a_{11}u_{1}u_{2}+a_{02}u_{2}^{2}$
is any polynomial of degree at most $2$ and
$\Phi_{S}(u_{1},u_{2})=l_{1}l_{2}l_{3}l_{4}l_{5}$, we obtain

$\begin{cases}w_{145}=-a_{10}-a_{20}-a_{11}\\\
w_{135}=-\frac{1}{2}(a_{11}-a_{02})\\\ w_{134}=\frac{1}{2}(a_{01}-a_{11})\\\
w_{125}=\frac{1}{4}(a_{00}-a_{10}-a_{11}+a_{02})\\\
w_{124}=\frac{1}{4}(a_{10}-a_{11})\\\
w_{123}=-\frac{1}{4}(a_{10}-a_{01}-a_{20}+a_{11}).\\\ \end{cases}$

Triangles $\Delta_{135}$ and $\Delta_{124}$ 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 $\mathfrak{F}^{\Delta}(S)$ 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
$\mathbf{v}_{1}=\mathbf{v}_{5}$. It shows that our technique can be
generalized to a certain class of multisets as well.

######  Example 4.

Let
$S=\\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3},\mathbf{v}_{4},\mathbf{v}_{5}\\}$
where
$\mathbf{v}_{1}=\mathbf{v}_{5}=(0,0),\mathbf{v}_{2}=(2,0),\mathbf{v}_{3}=(1,1),\mathbf{v}_{4}=(0,2)$.
Then $l_{1}=l_{5}=1,l_{2}=1-2u_{1},l_{3}=1-u_{1}-u_{2},l_{4}=1-2u_{2}$.
Calculating all products $l_{i}l_{j},\,i<j$ and taking their coefficients in
the standard monomial basis of $Pol(2,2)$, we obtain the following $6\times
10$-matrix $\widetilde{Mat}_{S}$:

$\widetilde{Mat}_{S}=\bordermatrix{~{}&l_{1}l_{2}&l_{1}l_{3}&l_{1}l_{4}&l_{1}l_{5}&l_{2}l_{3}&l_{2}l_{4}&l_{2}l_{5}&l_{3}l_{4}&l_{3}l_{5}&l_{4}l_{5}\cr
1&1&1&1&1&1&1&1&1&1&1\cr u_{1}&-2&-1&0&0&-3&-2&0&-1&-1&0\cr
u_{2}&0&-1&-2&0&-1&-2&-2&-3&-1&-2\cr u_{1}^{2}&0&0&0&0&2&0&0&0&0&0\cr
u_{1}u_{2}&0&0&0&0&2&4&0&2&0&0\cr u_{2}^{2}&0&0&0&0&0&0&0&2&0&0}.$

Its rank equals $6$ and a non-vanishing maximal minor consists of the columns
with numbers $\mathfrak{S}=\\{1,3,4,5,6,8\\}$. The corresponding submatrix
$\widetilde{Mat}_{S}(\mathfrak{S})$ equals

$\widetilde{Mat}_{S}(\mathfrak{S})=\bordermatrix{~{}&l_{1}l_{2}&l_{1}l_{4}&l_{1}l_{5}&l_{2}l_{3}&l_{2}l_{4}&l_{3}l_{4}\cr
1&1&1&1&1&1&1\cr u_{1}&-2&0&0&-3&-2&-1\cr u_{2}&0&-2&0&-1&-2&-3&\cr
u_{1}^{2}&0&0&0&2&0&0\cr u_{1}u_{2}&0&0&0&2&4&2\cr u_{2}^{2}&0&0&0&0&0&2}.$

Further,

$4\widetilde{Mat}_{S}^{-1}(\mathfrak{S})=\bordermatrix{~{}&1&u_{1}&u_{2}&u_{1}^{2}&u_{1}u_{2}&u_{2}^{2}\cr
l_{1}l_{2}&0&-2&0&-2&-1&0\cr l_{1}l_{4}&0&0&-2&0&-1&-2\cr
l_{1}l_{5}&4&2&2&1&1&1&\cr l_{2}l_{3}&0&0&0&2&0&0\cr
l_{2}l_{4}&0&0&0&-1&1&-1&\cr l_{3}l_{4}&0&0&0&0&0&2}.$

Thus, given an arbitrary rational function
$R(u_{1},u_{2})=P(u_{1},u_{2})/\Phi_{S}(u_{1},u_{2})$ where
$P(u_{1},u_{2})=a_{00}+a_{1,0}u_{1}+a_{0,1}u_{2}+a_{2,0}u_{1}^{2}+a_{11}u_{1}u_{2}+a_{02}u_{2}^{2}$
is any polynomial of degree at most $2$ and
$\Phi_{S}(u_{1},u_{2})=l_{1}l_{2}l_{3}l_{4}l_{5}$, we obtain

$\begin{cases}w_{345}=-\frac{1}{4}(2a_{10}+2a_{20}+a_{11})\\\
w_{235}=-\frac{1}{4}((2a_{01}+a_{11}+2a_{02})\\\
w_{234}=4a_{00}+2a_{01}+2a_{10}+a_{20}+a_{11}+a_{02}\\\ w_{145}=2a_{20}\\\
w_{135}=-a_{10}+a_{11}-a_{20}\\\ w_{125}=2a_{02}.\\\ \end{cases}$

Notice that triangles $\Delta_{125},\Delta_{135},\Delta_{145},\Delta_{234}$
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
$\mathfrak{F}^{\Delta}(S)$ in this example is given by the system of
equations:

$\begin{cases}a_{20}=0\\\ a_{10}=a_{11}\\\ a_{02}=0\\\
4a_{00}+2a_{01}+3a_{10}=0.\\\ \end{cases}$

##  5\. Remarks and open problems

######  Remark 15.

A weaker form of Corollary 4 (i.e. the rationality of
$F_{\mathcal{P}}^{\rho}(\mathbf{u})$, 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 [Bar2]*Lemma 1.

######  Problem 2.

Find an appropriate version of Theorem 2, applicable to non-simple and/or non-
convex polytopes.

######  Remark 16.

Choose an arbitrary basis $\\{\Delta_{j}\\}$ of $\mathfrak{M}^{\Delta}(S)$
consisting of the standard measures of simplices. The set $\\{\Delta_{j}\\}$
spans an integer lattice in $\mathfrak{M}^{\Delta}(S)$. (One can easily see
that this lattice is invariantly defined independently of the choice of a
basis of standard measures of simplices.) Denote by
$\mathfrak{M}_{\mathbb{Z}}^{\Delta}(S)$ the space $\mathfrak{M}^{\Delta}(S)$
with the latter lattice. We can prove the following.

######  Proposition 23.

If Conjecture 7 holds then any generalized polytope
$\mathcal{P}\in\mathcal{P}(S)$ with standard measure $\mu_{\mathcal{P}}$ in
$\mathfrak{M}^{\Delta}(S)$ corresponds to a rational point in
$\mathfrak{M}^{\Delta}_{\mathbb{Z}}(S)$.

###### Proof.

One can easily show that any $\mathcal{P}$ as above can be represented as the
union of the closures of connected components of $\mathbb{R}^{d}\setminus
H(S)$, where $H(S)$ is the hyperplane arrangement consisting of all
hyperplanes spanned by $d$-tuples of points in $S$. (The converse is obviously
not true.) Let $\tilde{S}\supseteq S$ be the extended set of vertices obtained
by adding to $S$ all vertices of the hyperplane arrangement $H(S)$. Since each
connected component in $\mathbb{R}^{d}\setminus H(S)$ is convex, it can be
triangulated on $\tilde{S}$. Consider the space
$\mathfrak{M}^{\Delta}_{\mathbb{Z}}(\tilde{S})$. Obviously,
$\mu_{\mathcal{P}}$ is an integer point in
$\mathfrak{M}^{\Delta}_{\mathbb{Z}}(\tilde{S})$. Also,
$\mathfrak{M}^{\Delta}_{\mathbb{Z}}(S)$ is contained in
$\mathfrak{M}^{\Delta}_{\mathbb{Z}}(\tilde{S})$ as a sublattice. Thus, if
$\mu_{\mathcal{P}}$ belongs to $\mathfrak{M}^{\Delta}_{\mathbb{Z}}(S)$ it is a
rational point there. ∎

######  Remark 17.

If either $\mathcal{P}$ itself or its complement in $\mathrm{conv}(S)$ can be
triangulated by simplices with vertices in $S$ then $\mu_{\mathcal{P}}$ is an
integer point in $\mathfrak{M}_{\mathbb{Z}}(S)$.

Let us pose the following tantalizing question.

######  Problem 3.

Does there exist a generalized polytope $\mathcal{P}$ with the set of vertices
$S(\mathcal{P})$ and with non-integer coordinates in
$\mathfrak{M}_{\mathbb{Z}}(S(\mathcal{P}))$?

######  Problem 4.

One can also define a rational convex cone
$\mathfrak{Pos}(S)\subset\mathfrak{M}_{\mathbb{Z}}(S)$ by taking non-negative
linear combinations of all $\mu_{\mathcal{P}}$, where $\mathcal{P}$ runs over
the set of all generalized polytopes in $\mathcal{P}(S)$.

######  Conjecture 24.

The rational cone $\mathfrak{Pos}(S)$ is uniquely determined by the oriented
matroid associated to $S$.

We conclude this section with the following question.

######  Problem 5.

Is it possible to describe the extremal rays of $\mathfrak{Pos}(S)$?

One can easily show that a simplex from $\mathcal{P}(S)$ spans an extremal ray
of $\mathfrak{Pos}(S)$ 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. [FuMiMo] and references therein.

## References