1 Introduction↩︎

The setting for this paper is the \(n\)-dimensional Euclidean space \(\mathbb{R}^n\). For two vectors \(x,y \in \mathbb{R}^n\) we denote by \(\langle x,y \rangle\) the standard scalar product of \(x\) and \(y\), and \(\| x \| = \sqrt{\langle x,x \rangle}\) denotes the associated Euclidean norm; \(\mathbb{S}^{n-1}=\{x\in\mathbb{R}^n : \| x \|=1\}\) is the \((n-1)\)-sphere. The convex hull of a non-empty set \(M\subset\mathbb{R}^n\) is denoted by \(\mathop{\mathrm{conv}}M\), and if \(M\) is finite then \(\mathop{\mathrm{conv}}M\) is called a polytope. By a result attributed to Minkowski and Weyl, \(P\subset\mathbb{R}^n\) is a polytope if and only if \[P=P(U,b)=\{x\in\mathbb{R}^n : U^\intercal x\leq b\}\] for a matrix \(U=(u_1,\dots,u_m)\in(\mathbb{S}^{n-1})^m\) with \(\mathop{\mathrm{pos}}U=\mathbb{R}^n\) and \(b\in\mathbb{R}^m\). Here \(\mathop{\mathrm{pos}}U\) means the positive hull, i.e., the set of all non-negative linear combinations of the column vectors \(u_1,\dots,u_m\in\mathbb{S}^{n-1}\) of \(U\). Apparently, we may assume that the column vectors are pairwise different, and therefore we set \[\mathcal{U}(n,m)=\left \{U=(u_1,\dots,u_m)\in(\mathbb{S}^{n-1})^m : \mathop{\mathrm{pos}}U=\mathbb{R}^n, u_i\ne u_j, i \ne j\right\}.\]

For \(1\leq i\leq m\) let \[F_i(b)=F(u_i,b)=P\cap\{x\in\mathbb{R}^n : \langle u_i,x\rangle =b_i\}\] which is always a face of \(P\), and, of course, might be empty. If \(\dim F_i(b)=\dim P-1\), \(F_i(b)\) is called a facet of \(P\). For \(M\subset \mathbb{R}^n\) we denote by \(\mathrm{vol}\,(M)\) its volume, i.e., its \(n\)-dimensional Lebesgue measure. If \(M\) is contained in a \(k\)-dimensional plane \(A\), \(\mathrm{vol}\,_k(M)\) refers to the \(k\)-dimensional Lebesgue measure with respect to \(A\).

We will mainly assume that \(b\geq 0\). This implies \(0\in P\), and if \(b > 0\) then \(0 \in \mathop{\mathrm{int}}P\), i.e., \(0\) is an interior point of \(P\), and so \(\dim P=n\). If \(F_i(b)\) is a facet of \(P(U,b)\), \(\dim P(U,b)=n\), then \(\frac{1}{n}b_i\mathrm{vol}\,_{n-1}(F_i(b))\) is the volume of the cone (pyramid) \(\mathop{\mathrm{conv}}(\{0\}\cup F_i(b))\). For \(U\in\mathcal{U}(n,m)\), the polytope \(P(U,b)\) is the interior-disjoint union of all these cones, so we can write \[\mathrm{vol}\,(P(U,b))=\frac{1}{n}\sum_{i=1}^m b_i\mathrm{vol}\,_{n-1}(F_i(b)).\] For such a \(P=P(U,b)\), \(\dim P=n\), we consider its cone-volume measure \(\mathop{\mathrm{V}}_P\) which is the finite non-negative Borel measure \(\mathop{\mathrm{V}}_P:\mathcal{B}(\mathbb{S}^{n-1})\to\mathbb{R}_{\geq 0}\) given by \[\mathop{\mathrm{V}}_P(\eta) =\sum_{i=1}^m \frac{b_i}{n}\mathrm{vol}\,_{n-1}(F_i(b))\,\delta_{u_i}(\eta) =\sum_{u_i\in\eta} \frac{b_i}{n}\mathrm{vol}\,_{n-1}(F_i(b)).\] Here \(\eta\subseteq \mathbb{S}^{n-1}\) is a Borel set and \(\delta_{u_i}(\cdot)\) is the Dirac measure in \(u_i\), i.e., \(\delta_{u_i}(\eta)=1\) if \(u_i\in\eta\), otherwise it is \(0\).

The discrete logarithmic Minkowski (existence) problem introduced by Böröczky, Lutwak, Yang and Zhang [1] asks for necessary and sufficient conditions such that a finite discrete Borel measure \[\mu:\mathcal{B}(\mathbb{S}^{n-1})\to\mathbb{R}_{\geq 0}, \quad \mu(\eta)=\sum_{i=1}^m \gamma_i\,\delta_{u_i}(\eta) \label{eq:measure}\tag{1}\] with \(u_i\in\mathbb{S}^{n-1}\), \(\gamma_i> 0\), is the cone-volume measure of a polytope. We will denote such a measure also by \(\mu(U,\gamma)\), where \(\gamma\in\mathbb{R}^m_{>0}\) is the vector with entries \(\gamma_i\).

This discrete problem can be extended to the continuous setting, i.e., to the space of all convex bodies and the corresponding general logarithmic Minkowski problem is a cornerstone of modern convex geometry. The associated partial differential equation for the logarithmic Minkowski problem is the following Monge-Ampère type equation on the unit sphere: For a given function \(f: \mathbb{S}^{n-1} \rightarrow (0,\infty)\), solve for the support function \(h: \mathbb{S}^{n-1} \rightarrow (0,\infty)\) of a convex body, \[h \det(h_{ij} + h \delta_{ij}) = f,\] where \(h_{ij}\) is the covariant derivative of \(h\) with respect to an orthonormal frame on \(\mathbb{S}^{n-1}\) and \(\delta_{ij}\) is the Kronecker delta. The Monge-Ampère equation has a close relation to the optimal transport with quadratic cost [2].

The cone-volumes are instrumental for computing Wachspress coordinates [3], which define the adjoint of a polytope and thereby its canonical form. The canonical form was introduced in the context of positive geometries and scattering amplitudes in quantum field theory [4]. The adjoint of a polytope appears in many different mathematical contexts (cf. [5]) and is relevant for convex optimization [6].

The cone-volume measure has found numerous important applications in convex geometry and analysis. In particular, its properties have been used to establish reverse affine isoperimetric inequalities [7], [8]. Since the work of Gromov and Milman [9], it has become a central tool, with further applications to functional inequalities, asymptotic geometric analysis, and probability theory [10]–[12]. For its history, relevance and impact we refer to [1], [13]–[20] and to the references within. Here we will only focus on the discrete setting.

The subspace concentration condition (scc), introduced by Böröczky et al. [1], plays an important role in the classification of the cone-volume measure. A finite discrete Borel measure \(\mu=\mu(U,\gamma): \mathcal{B}(\mathbb{S}^{n-1})\to\mathbb{R}_{\geq 0}\) with \(U\in\mathcal{U}(n,m)\), \(\gamma >0\), is said to satisfy the scc  if

1. for every proper linear subspace \(L\subset\mathbb{R}^n\) it holds \[\mu(L)= \sum_{u_i\in L} \gamma_i \leq \frac{\dim L}{n}\mu(\mathbb{S}^{n-1}), \label{eq:scc1}\tag{2}\]

2. and equality holds in 2 if and only if there exists a subspace \(\overline{L}\) complementary to \(L\) such that \(\{u_1,\dots,u_m\}\subset L\cup\overline{L}\).

In Section 2 we will define for \(U\in\mathcal{U}(n,m)\) the polytope \(P_{{\tt scc}}(U)\) (see 5 ), which we call the subspace concentration polytope (of \(U\)) that captures the scc. Up to scaling the polytope \(P_{\tt scc}(U)\) is (just) the matroid base polytope of the set of column vectors of \(U\).

Here \(\mathop{\mathrm{relint}}(M)\) denotes the relative interior of \(M\subseteq\mathbb{R}^n\), i.e., the sets of interior points with respect to the ambient space given by \(\mathop{\mathrm{aff}}M\), the affine hull of \(M\).

In the special case that \(U\) does not contain parallel vectors, the polytope \(P_{\tt scc}((U,-U))\) as well as Proposition 1.1 with the additional symmetry assumption \(\gamma_i=\gamma_{m+i}\), \(1\leq i\leq m\), was already considered by Liu et al. [21].

In order to show the relation of the scc  to the cone-volume measure we also define a cone-volume set \(C_{\tt cv}(U)\). To this end, we firstly consider for \(U\in\mathcal{U}(n,m)\) and \(b\in \mathbb{R}^m_{\geq 0}\) the cone-volume vector \[\gamma(U,b)=\frac{1}{n}\Big(b_1\mathrm{vol}\,_{n-1}(F(u_1)),\dots, b_m\mathrm{vol}\,_{n-1}(F(u_m))\Big)^\intercal\in \mathbb{R}^m_{\geq 0}.\] Observe that some of its entries might be zero, if \(F_i(b)\) is not of dimension \(n-1\) or \(b_i=0\). The set \[C_{\tt cv}(U)=\left\{ \gamma(U,b) : b\in\mathbb{R}^m_{\geq 0} \text{ and } \mathrm{vol}\,(P(U,b))=1\right\}\] is called the cone-volume set of \(U\). Any cone-volume vector \(\gamma(U,b)\) of an \(n\)-dimensional polytope of the type \(P(U,b)\) is up to scaling to volume 1 an element of \(C_{\tt cv}(U)\) as \[\frac{1}{\mathrm{vol}\,(P(U,b))}\gamma(U,b) = \gamma\left(U, \left(\mathrm{vol}\,(P(U,b)) \right)^{-1/n}b\right)\in C_{\tt cv}(U). \label{eq:scaling}\tag{3}\]

If \(m=2m'\) is even and \(u_{m'+i }=-u_i\), \(i=1,\dots,m'\), we denote such a matrix by \(U^s\in \mathcal{U}(n,m)\) and a vector \(b\in\mathbb{R}^m\) satisfying \(b_i=b_{m'+i}\), \(i=1,\dots,m'\), will be denoted by \(b^s\). Let \[C^s_{\tt cv}(U^s)=\left\{ \gamma(U^s,b^s) : b^s\in\mathbb{R}^m_{\geq 0}, \mathrm{vol}\,(P(U^s,b^s))=1\right\}\] be the associated symmetric cone-volume set. In the groundbreaking paper [1] it was in particular shown that an even finite positive Borel measure \(\mu(U^s,\gamma^s)\) is the cone-volume measure of an origin symmetric polytope \(P(U^s,b^s)\) if and only if \(\mu(U^s,\gamma^s)\) satisfies scc. Hence, with Proposition 1 this can be reformulated as (see also [21])

In the general setting we will show that \(C_{\tt cv}(U)\) and \(P_{\tt scc}(U)\) coincide only for parallelepipeds.

For the (non-symmetric) discrete logarithmic Minkowski problem we do not know necessary and sufficient conditions. By a result of Chen et al. [20], however, we have the following inclusion.

In Section 2 we will also see that both sets have the same dimension (Proposition 6 and Proposition 7).

By definition, for \(U\in\mathcal{U}(n,m)\), the cone-volume set \(C_{\tt cv}(U)\) is a subset of \(\{x\in\mathbb{R}^{m}: x\geq 0, \, x_1+x_2+\dots + x_m=1\}\). A result of Zhu shows that it can also be that large.

In general the inclusion in Theorem 2 is strict as \(C_{\tt cv}(U)\) might not be convex and even not representable as the finite union of polytopes (see Section 2). Our main result is that \(C_{\tt cv}(U)\) is (at least) a semialgebraic set, i.e., roughly speaking, it can be described by the finite union of sets which are representable by finitely many polynomial inequalities.

In the special case \(n=2\), this was shown already by Stancu [22], explicit descriptions of \(C_{\tt cv}(U)\) for planar quadrilaterals were obtained by Liu et al. [23], where the trapezoid case was already studied by Pollehn [24]. In addition, a general valid polynomial inequality for arbitrary \(C_{\tt cv}(U)\) was obtained by Böröczky and Hegedűs [25]. Representations related to particular higher dimensional convex bodies were recently studied by Chen, Liu and Xiong (private communication).

Our general polynomial description reduces to Stancus representation in the planar case. We will also present a bound on the degree of the polynomials in the general case (see Corollary 2).

The paper is organized as follows. In Section 2 we will define \(P_{scc}(U)\), give a proof of Proposition 1, show the relation to matroid polytopes, study certain basic properties of the two sets \(P_{\tt scc}(U)\) and \(C_{\tt cv}(U)\), and also provide a few examples. In particular, we will also show that \(C_{\tt cv}(U)\) is path-connected (see Proposition 9) and we will provide the proof of Theorem 2. The proof of Theorem 4 is given in Section 3 where we will also present some necessary background on semialgebraic sets. Section 4 deals with the 2-dimensional case, and in Section 5 we wil briefly discuss the non-uniqueness of cone-volume vectors.

2 Subspace concentration polytopes and cone-volume sets↩︎

In the following let \(U\in\mathcal{U}(n,m)\). With \(S\subseteq U\) we mean a subset of the column vectors, and \(\mathop{\mathrm{rg}}(S)\) denotes the rang of the matrix \(S\), i.e., \(\dim(\mathop{\mathrm{lin}}S)\). We will treat \(S\subseteq U\) as matrix as well as the set consisting of its column vectors.

Let \(\mathcal{B}(U)\) denotes all subsets of \(U\) forming a basis of \(\mathbb{R}^n\), then the tuple \(M_U=(U, \mathcal{B}(U))\) is called the basis matroid of \(U\). The associated characteristic polytope \[P(M_U) =\mathop{\mathrm{conv}}\left\{ \chi_U(B) : B\in \mathcal{B}(U) \right\}\subset\mathbb{R}^m\] is called the (basis) matroid polytope of \(M_U\). Here \(\chi_U(B)\in\mathbb{R}^m\) is the characteristic vector of the basis \(B\) with respect to \(U\), i.e., for \(1\leq i\leq m\) its \(i\)th entry is \(1\) if column \(u_i\in B\), otherwise \(0\). For general information on matroids we refer to [26], [27]. It is well-known that \(P(M_U)\) can also be described by the following system of inequalities (see, e.g., [28]) \[\begin{align} P(M_U) & = \left\{x\in\mathbb{R}^m : x\geq 0,\, \sum_{i=1}^m x_i=n, \sum_{u_i\in S} x_i\leq \mathop{\mathrm{rg}}(S) \text{ for all } S\in \mathcal{L}(U)\right\}, \end{align}\] where \[\begin{align} \mathcal{L}(U) & = \{ S \subseteq U : 1 \leq \mathop{\mathrm{rg}}(S) \leq n - 1 \text{ and } U \cap \mathop{\mathrm{lin}}S = S \}. \end{align}\] The subsets in \(\mathcal{L}(U)\) are called flats, and for a flat \(S\) the associated rank inequality is an (implicit) equality for \(P(M_U)\) if and only if \(S\) belongs to the set \[\mathcal{F}(U) = \{ S \in \mathcal{L}(U) : \mathop{\mathrm{lin}}(S) \cap \mathop{\mathrm{lin}}(U \setminus S) = \{ 0 \} \},\] which are the so-called (non-trivial) separators of the matroid (see, e.g., [27], [29]). For later purpose and in view of scc ii) we note that \[\begin{align} \{\mathop{\mathrm{lin}}S : S \in \mathcal{F}(U) \} = \{ & L : L\subset\mathbb{R}^n \text{ is a proper subspace such that } \\[-1ex] &\text{ there exits a complementary}\\[-1ex] &\text{ subspace }L'\text{ with } U\subset L\cup L' \}. \end{align} \label{eq:second}\tag{4}\] With these two sets we define the subspace concentration polytope as the base matroid polytope scaled by \(1/n\): \[\begin{align} P_{\tt scc}(U)= \frac{1}{n} P(M_U) \\ =\Biggl\{ x \in \mathbb{R}^m & : \sum_{i = 1}^m x_i = 1, \sum_{u_i \in S } x_i = \frac{\mathop{\mathrm{rg}}(S)}{n}, S \in \mathcal{F}(U), \\ &\, x\geq 0, \sum_{u_i \in S } x_i \leq \frac{\mathop{\mathrm{rg}}(S)}{n},\, S \in \mathcal{L}(U) \setminus \mathcal{F}(U) \Biggl\}. \end{align} \label{def:pscc}\tag{5}\]

Next we remark that \(P_{\tt scc}(U)\) as well as \(C_{\tt cv}(U)\) are linear invariant which will be used later on.

For the proof of Proposition 1 it will be convenient first to give an explicit description of \(\mathop{\mathrm{relint}}P_{\tt scc}(U)\). It immediately follows from the above mentioned role of the separators but for completness sake we add the short proof.

Now we are ready to prove that \(\mathop{\mathrm{relint}}P_{\tt scc}(U)\) describes the subspace concentration conditions.

Before we proceed we have to extend the definitions of \(P(U,b)\), \(C_{\tt cv}(U)\) and \(P_{\tt scc}(U)\) to subsets \(S\subseteq U\) with \(\mathop{\mathrm{pos}}S=\mathop{\mathrm{lin}}S\). We will do this always with respect to the “ambient matrix” \(U\), i.e., for a vector \(v\in\mathbb{R}^{|U|}\) we denote by \(v_S\in\mathbb{R}^{|S|}\) the subvector of \(v\) having coordinates \(v_i\) with \(u_i\in S\). Then with \(\mathcal{B}(S)=\{T\subseteq S: T \text{ basis of }\mathop{\mathrm{lin}}S\}\) we set \[\begin{align} P_{\tt scc}(S) &=\mathop{\mathrm{conv}}\{\chi_U(T) : T\in \mathcal{B}(S) \}. \end{align}\] With the canonical definitions of \(\mathcal{L}(S)\), \(\mathcal{F}(S)\) we have \[\begin{align} P_{\tt scc}(S) & =\Bigg \{ x \in\mathbb{R}^{|U|} : x_{U\setminus S}=0, x_S\geq 0, \sum_{u_i\in S}x_i=1,\\ &\quad\quad\quad \quad\quad\quad\sum_{u_i\in T} x_i\leq \frac{\mathop{\mathrm{rg}}T}{\mathop{\mathrm{rg}}S} \text{ for } T\in\mathcal{L}(S),\\ &\quad\quad\quad \quad\quad\quad \sum_{u_i\in T} x_i= \frac{\mathop{\mathrm{rg}}T}{\mathop{\mathrm{rg}}S} \text{ for } T\in\mathcal{F}(S) \Bigg \}. \end{align}\] Regarding cone-volume sets let \[P(S,b_S)=\{x\in\mathop{\mathrm{lin}}S: S^\intercal x\leq b_s\}.\] Let now \(\gamma=\gamma(S,b_s)\in\mathbb{R}^{|U|}\) be the cone-volume vector with \(\gamma_{U\setminus S}=0\) and for \(u_i\in S\) let \(\gamma_i\) be the associated cone-volume of the \(\mathop{\mathrm{rg}}(S)\)-dimensional polytope \(P(S,b_s)\), i.e., \[\gamma_i=\frac{b_i}{\mathop{\mathrm{rg}}S}\mathrm{vol}\,_{\mathop{\mathrm{rg}}(S)-1}(F_S(u_i)),\] where \(F_s(u_i)=P(S,b_s)\cap\{x\in\mathop{\mathrm{lin}}S : \langle u_i,x\rangle =b_i\}\). Then we set \[C_{\tt cv}(S)=\{\gamma(S,b_s) : b\in\mathbb{R}^{|U|}_{\geq 0} \text{ and }\mathrm{vol}\,_{\mathop{\mathrm{rg}}S}(P(S,b_s))=1\}.\]

Observe that for \(S\in\mathcal{F}(U)\) we always have \(\mathop{\mathrm{pos}}S=\mathop{\mathrm{lin}}S\), and with the separators we can write \(P_{\tt scc}(U)\) as direct sum of submatroid polytopes. In fact, given \(S\in\mathcal{F}(U)\) it is known (e.g., [27]) that \[P_{\tt scc}(U)=\frac{\mathop{\mathrm{rg}}(S)}{n}P_{\tt scc}(S)\oplus \frac{\mathop{\mathrm{rg}}(U\setminus S)}{n}P_{\tt scc}(U\setminus S). \label{eq:sepsplit}\tag{9}\] Iterating this process, i.e, looking at separators of \(S\) and \(U\setminus S\) and so forth leads to a unique partion \[U=S_1\cup S_2 \cup \cdots \cup S_d \label{eq:uniquepart}\tag{10}\] into so-called irreducible sets \(S_j\subseteq U\), i.e., \(\mathcal{F}(S_j)=\varnothing\), \(1\leq j\leq d\). In particular, we have \[\mathbb{R}^n =\mathop{\mathrm{lin}}S_1\oplus \dots \oplus \mathop{\mathrm{lin}}S_d.\]

Next we present three examples.

We further remark that the vertices of \(P_{\tt scc}(U)\) are exactly the vectors \(\chi_U(B)\), \(B\in\mathcal{B}\), and all edges are parallel to a vector of the type \(e_i-e_j\). [28]

As well-understood as \(P_{\tt scc}(U)\) is, the cone-volume set \(C_{\tt cv}(U)\) is equally mysterious is (in general). But first we note that we also have a decomposition as in Proposition 6.

By Proposition 7, Proposition 6 and Theorem 2 we have \[\mathop{\mathrm{aff}}C_{\tt cv}(U)=\mathop{\mathrm{aff}}P_{\tt scc}(U)\] and thus we get

Example 1 (Polytopes in general positions, e.g., simplex)continued. Theorem 3 shows that for \(U=(u_1,\dots,u_m)\) in general position we have \[C_{\tt cv}(U)\cap\mathbb{R}^m_{>0}=\mathop{\mathrm{relint}}\mathop{\mathrm{conv}}\{e_1,\dots,e_m\}. \label{eq:general}\tag{13}\] For a simplex, i.e., \(m=n+1\), this is easy to see. To this end we observe that by the existence theorem of Minkowski (see, e.g., [30]) there exists an unique – up to translations – simplex \[T(b^*)=\{ x\in\mathbb{R}^n : \langle u_i, x\rangle\leq b_i^*, 1\leq i\leq n+1\}\] with \(\mathrm{vol}\,(T)=1\). Let \(\phi_i\) be the \((n-1)\)-dimensional volume of the facet with outer unit normal vector \(u_i\). For \(\gamma=(\gamma_1,\dots,\gamma_{n+1})^\intercal \in\mathop{\mathrm{conv}}\{e_1,\dots, e_{n+1}\}\), let \(b_i=n\,\gamma_i/\phi_i\), \(1\leq i\leq n+1\). Then \(T(b)\) is a simplex containing the origin, with \(\gamma(U,b)=\gamma\), and thus \(\mathrm{vol}\,(T)=1\). As we always have \(C_{\tt cv}(U)\subseteq \mathop{\mathrm{conv}}\{e_1,\dots,e_m\}\) this gives 13 for \(m=n+1\). \(\triangle\)

Example 2 (Parallelepiped)continued. Let \(S_j=\{u_j,u_{n+j}\}=\{u_j,-u_j\}\), \(j=1,\dots,n\), be the irreducible sets of \(U^s\). Then \[C_{\tt cv}(S_j)=\mathop{\mathrm{conv}}\{e_j,e_{n+j}\},\] and with Proposition 7 we obtain \(C_{\tt cv}(U^s)=P_{\tt scc}(U^s)\). Note that we consider the general cone-volume set without restricting to the symmetric case (see Theorem 2). \(\triangle\)

Example 3 (Trapezoid)continued. From [24] we get that \[\begin{align} C_{\tt cv}(U)&\cap\mathbb{R}^4_{>0} = \left\{ \gamma \in \mathbb{R}_{> 0}^4: \sum_{i=1}^4 \gamma_i = 1, \gamma_1 + \gamma_3 < \gamma_2 + \gamma_4 \right\} \\ & \cup \left\{ \gamma \in \mathbb{R}_{> 0}^4: \sum_{i=1}^4 \gamma_i = 1, \gamma_1 + \gamma_3 \geq \gamma_2 + \gamma_4 \geq 2 \sqrt{\gamma_1 \gamma_3} \text{ and } \gamma_1 < \gamma_3 \right\}. \end{align} \label{eq:trapeziod-hannes}\tag{14}\] \(\triangle\)

Trapezoids also serve as an example in order to show the following properties of \(C_{\tt cv}(U)\).

Proof. We will only present \(2\)-dimensional examples, which can, however, easily be extended to any dimension via Proposition 7.

For i) and ii) we take the trapezoid from Example 3, and let \(A\) be the first set of the (disjoint) union 14 and \(B\) the second one. Obviously, \(B\) is not convex and Figure 1 presents a visualization.

For ii) let \(\gamma=(1/9,2/9,4/9,2/9)^\intercal\). Then \[\gamma_1+\gamma_3>\gamma_2+\gamma_4=2\sqrt{\gamma_1\,\gamma_3}, \gamma_1<\gamma_3, \text{ and } \gamma_1+\gamma_2+\gamma_3+\gamma_4=1,\] and by 14 we have \(\gamma\in C_{\tt cv}(U) \cap\mathbb{R}^m_{>0}\). However, for (small) \(\epsilon>0\) the vector \(\overline{\gamma}=(1/9+\epsilon,2/9-\epsilon,4/9,2/9)^\intercal\) still satisfies all the above strict inequalities but as \(\overline{\gamma}_2+\overline{\gamma}_4<2\sqrt{\overline{\gamma}_1\,\overline{\gamma}_3}\) it is not contained in \(C_{\tt cv}(U)\). Hence, \(\gamma\in\mathop{\mathrm{bd}}C_{\tt cv}(U)\).

To see iii), i.e., \(C_{\tt cv}(U)\) is not closed in general we consider for \(U=(e_2,e_1+e_2,-e_2,-e_1+e_2)\) the trapezoids \(P_U(b_\epsilon)\) with \(b_\epsilon=(\epsilon,0,0,1/\epsilon+\epsilon)^\intercal\).

By elementary calculations we get for the cone-volume vector \(\gamma(\epsilon)=\gamma((U,b_\epsilon)\) \[\gamma(\epsilon)_1=\frac{1}{2}\left(1-\frac{\epsilon^2}{2}\right), \gamma(\epsilon)_2=\gamma(\epsilon)_3=0, \gamma(\epsilon)_4=\frac{1}{2}\left(1+\frac{\epsilon^2}{2}\right).\] Hence, \((1/2,0,0,1/2)^\intercal\in\mathop{\mathrm{cl}}(C_{\tt cv}(U))\) but apparently there is no right hand side \(b\) such that the trapezoid \(P_U(b)\) has this cone-volume vector. ◻

We remark that for \(U\in\mathcal{U}(n,m)\) we always have \[\mathop{\mathrm{relint}}C_{\tt cv}(U) \subset C_{\tt cv}(U)\cap\mathbb{R}^m_{>0}.\]

Next we point out that \(C_{\tt cv}(U)\) is path-connected.

In the example of a parallelepiped we have seen that for \(U^s\in\mathcal{U}(n,2n)\) we have \(C_{\tt cv}(U)=P_{\tt scc}(U)\). This is also essentially the only case as claimed in Theorem 2.

For later purpose we also point out that the cone-volume vectors are continuous.

Finally, we remark that it is also possible to consider instead of the polytope \(P_{\tt scc}(U)\) the half open subspace concentration set \[\begin{align} \widehat{P_{\tt scc}} (U)= \Biggl\{ x \in \mathbb{R}^m & : \sum_{i = 1}^m x_i = 1, \sum_{u_i \in S } x_i = \frac{\mathop{\mathrm{rg}}(S)}{n}, S \in \mathcal{F}(U), \\ &\, x\geq 0, \sum_{u_i \in S } x_i < \frac{\mathop{\mathrm{rg}}(S)}{n},\, S \in \mathcal{L}(U) \setminus \mathcal{F}(U) \Biggl\}. \end{align}\] Then it can be shown that relations in Theorem 1 and Theorem 2 become \[\begin{align} C^s_{\tt cv}(U^s) & =\widehat{P_{\tt scc}} (U^s) \cap \{ x \in \mathbb{R}^m : x_{i } = x_{i + m'},\,1\leq i\leq m' \}, \\ \quad C_{\tt cv}(U)&\supseteq \widehat{P_{\tt scc}}(U). \end{align}\] However, we prefer to work with the closed polytope \(P_{\tt scc}(U)\).

3 Semialgebraic Sets and Cone-Volumes↩︎

First we recall that a semialgebraic set in \(\mathbb{R}^m\) is the finite union of sets of the form \[\{ x\in\mathbb{R}^m : f_i(x)\geq 0, i\in I, g_j(x)>0, j\in J \}\] where \(I,J\subseteq \mathbb{N}\) are finite and \(f_i,g_j\in\mathbb{R}[x]\) are polynomials. By definition, the finite union of semialgebraic sets is semialgebraic and by the classical Tarski-Seidenberg principle the projection of a semialgebraic set is again a semialgebraic set, see [33].

In order to show that \(C_{\tt cv}(U)\) is a semialgebraic set, we will first focus on the cone-volume vectors of \(n\)-polytopes \(P(U,b)\) which are simple and strongly isomorphic. An \(n\)-dimensional polytope is called simple if each vertex is contained in exactly \(n\) facets, and two polytopes are strongly isomorphic if their face lattice is isomorphic and the affine hulls of the facets are parallel (cf. [30]). In our setting this implies that if for \(b,\overline{b}\in\mathbb{R}^m_{\geq 0}\) two \(n\)-polytopes \(P(U,b)\) and \(P(U,\overline{b})\) are combinatorially isomorphic, i.e., their face lattices are isomorphic, then they are also strongly isomorphic.

Observe, for a”generic” right hand side vector \(b\in\mathbb{R}^m_{\geq 0}\) the polytope \(P(U,b)\) is simple and the general case will be deduced from the simple case by approximation. The next lemma collects some well-known properties of strongly isomorphic simple polytopes.

The cones \(A_1(U),\dots,A_l(U)\) are called the type-cones of \(U\), cf. [34], and next we investigate them for our running examples.

Example 1 (Simplex)continued. Let \(U \in\mathcal{U}(n,n+1)\) be in general position. Then every \(b\in\mathbb{R}^m_{\geq 0}\), \(b\ne 0\), \(P(U,b)\) is an \(n\)-dimensional simplex. Thus there is only one type cone \(A_1(U)=\mathbb{R}^m_{\geq 0}\), and the volume of \(P(U,b)\) or of its facets can easily be calculated via determinants, which then are polynomials in the coordinates of \(b\). \(\triangle\)

Example 2 (Parallelepiped) continued. Here again we have only one type-cone \(A_1(U^s) = \mathbb{R}_{\geq 0}^m\). This follows from the fact that two \(n\)-polytopes \(P(U^s,b)\) and \(P(U^s,\tilde{b})\), where \(b, \tilde{b} \in\mathbb{R}^m_{\geq 0}\) are \(n\)-dimensional parallelepipeds with the same facet directions. The volume polynomial is given by \[\mathrm{vol}\,(P(U^s,b^s))= \frac{ \prod_{i=1}^n (b_i+b_{n+i})}{ |\det(A)|},\] where \(A\subset U^s\) consists of the first \(n\) columns. \(\triangle\)

Example 3 (Trapezoid) continued. We assume that the vectors in \(U\) are ordered counter-clockwise, and \(u_1 = e_2= - u_3\), see Figure 2.

Let \(a_2, a_4 \in \mathbb{R}\), \(a_4 > 0 > a_2\) such that \(u_2 = \frac{1}{l_2}(-1,-a_2)^T\) and \(u_4 = \frac{1}{l_4}(1,a_4)^T\), with \(l_2 = \| (-1,-a_2) \|\) and \(l_4 = \| (1,a_4) \|\). For \(b \in \mathbb{R}_{>0}\), the polygon \(P = P(U,b)\) is a \(2\)-dimensional trapezoid if and only if the intersection point of the lines \(H(u_2,b_2)\) and \(H(u_4,b_4)\) is above the line \(H(u_1,b_1)\), where \(H(u_i,b_i)=\{x\in\mathbb{R}^2 : \langle u_i,x\rangle=b_i\}\). This intersection point is given by \[\frac{1}{a_4-a_2} \begin{bmatrix} -(l_2 a_4 b_2 + l_4 a_2 b_4) \\ l_2 b_2 + l_4 b_4 \end{bmatrix}\] and so we have a \(2\)-dimensional trapezoid if and only if \(l_2 b_2 + l_4 b_4 > (a_4 - a_2) b_1\). Thus, the hyperplane separating the two different type-cones is given by \[H= \{ b \in \mathbb{R}_{\geq 0}^4 : l_2 b_2 + l_4 b_4 - (a_4 - a_2) b_1 = 0 \}\] and the two different type-cones are \[\begin{align} A_1(U) & = \{ b \in \mathbb{R}_{\geq 0}^4 : l_2 b_2 + l_4 b_4 - (a_4 - a_2) b_1 \geq 0 \} \text{ and } \\ A_2(U) & = \{ b \in \mathbb{R}_{\geq 0}^4 : l_2 b_2 + l_4 b_4 - (a_4 - a_2) b_1 \leq 0 \}. \end{align}\] For \(b\in \mathop{\mathrm{int}}A_1(U)\) we get \(2\)-dimensional trapezoids, and \(2\)-dimensional triangles for \(b\in \mathop{\mathrm{int}}A_2(U)\) as well as for \(b\in\mathbb{R}^4_{>0}\cap H\). \(\triangle\)

In the following, we assume that for \(U\in\mathcal{U}(n,m)\) and \(k\in\{1,\cdots,l\}\) the polyhedral cone \(A_k(U)\) from Lemma 3 is given by \[A_k(U)=\{b\in\mathbb{R}^m: B_k \cdot b\geq 0\}\] for some matrix \(B_k\in\mathbb{R}^{m_k\times m}\). For the computation of such a matrix \(B_k\), we refer to [35]. In view of Lemma 3 we set for \(1\leq k\leq l\) \[W_k(U)=\left\{ (\gamma,b) \in\mathbb{R}^m \times \mathbb{R}^m : B_k\,b >0, v_k(b)=1, \gamma_i= \frac{f_{k,i}(b)\cdot b_i}{n}, 1\leq i\leq m\right\}. \label{eq:conevolumevectorstype}\tag{17}\] Observe that for \((\gamma,b)\in W_k(U)\), we have \(b\in\mathop{\mathrm{int}}A_k(U)\) and so we know by Lemma 3 that \(P(U,b)\) is a simple polytope with cone-volume vector \[\gamma(U,b)=\gamma.\] Apparently, \(W_k(U)\) is a semi-algebraic set and they are the main ingredients of the proof of Theorem 4 which will immediately follow from the next lemma.

Apparently, Lemma 4 implies Theorem 4.

Example 1 (Simplex) continued. By the existence theorem of Minkowski all simplices \(P(U,b)\) of volume \(1\) are translates of each other. With \(\alpha_i=\mathrm{vol}\,(P(U,e_i))\), \(1\leq i\leq n+1\), we conclude \[\mathop{\mathrm{cl}}W_1(U) = \mathop{\mathrm{conv}}\left\{(e_i, \alpha_i^{-1/n}e_i): 1\leq i\leq n+1\right\}.\] \(\triangle\)

So in this case \(\mathop{\mathrm{cl}}W_1(U)\) is convex, which is, of course, not true in general, as already a parallelepiped shows.

Example 2 (Parallelepiped) continued. We have seen already that the first \(n\) coordinates of \(\mathop{\mathrm{cl}}(W_1(U^s))\) are convex, in the sense that its projection onto the first \(n\) coordinates maps the set \(\mathop{\mathrm{cl}}(W_1(U^s))\) onto \(C_{{\tt cv}}(U^s)\) which is equal to \(P_{\tt scc}(U^s)\). However, the last \(n\) coordinates do not behave convex: take two right hand sides \(b,\tilde{b}\in\mathbb{R}^{2n}_{\geq 0}\) such that the \(n\)-parallelepipeds \(P(U,b)\), \(P(U,\tilde{b})\) have volume \(1\) but are not homothetic. Then by the Brunn-Minkowski theorem \(P(U, \frac{1}{2}b+\frac{1}{2}\tilde{b})\) has volume greater than 1. \(\triangle\)

Generalizing the observation made by the parallelepiped, we obtain the following characterization.

In order to improve the representation of the \(C_{\tt cv}(U)\) as a projection of the sets \(\cup_{k=1}^l\mathop{\mathrm{cl}}(W_k(U))\) let us assume that \(\mathcal{I}(U)\subseteq\{1,\dots,l\}\) be all indices such for \(j\in \mathcal{I}(U)\) and \(b\in\mathop{\mathrm{int}}A_j(U)\), \(F(u_i,b)\) is a facet of \(P(U,b)\) for all \(1\leq i\leq m\). In words, \(\mathcal{I}(U)\) represents all the type cones \(A_k(U)\) where for an interior vector \(b\) all vectors \(u_i\), \(1\leq i\leq m\), are outer unit normal vectors of facets of \(P(U,b)\).

4 Polynomial Inequalities for Polygons↩︎

In this section, we present for \(U=(u_1,\dots,u_m)\in\mathcal{U}(2,m)\) a set of polynomials describing the set \(W_k(U)\). In view of Corollary 12 we will do it only for \(k\in\mathcal{I}(U)\), i.e., we consider only type-cones \(A_k(U)\) such that \(F(u_i,b)\), \(1\leq i\leq m\), are facets (edges) for all \(b\in\mathop{\mathrm{int}}A_k(U)\). Since \(n=2\) there exists only one such type-cone, which we will denote by \(A(U)\) and the associated set with the cone-volume vectors will be denoted by \(W(U)\) (cf. 17 ).

In order to describe \(A(U)\) and \(W(U)\) we assume that the unit vectors \(u_1,\dots,u_m\) are ordered counter-clockwise. Now \(u_1,\dots,u_m\) are the outer unit normal vectors of edges of \(P(U,b)\) if and only if the intersection point \(v_l\) of the two (neighbouring) lines \(\{x\in\mathbb{R}^2 : \langle u_{l},x\rangle=b_{l}\}\) and \(\{x\in\mathbb{R}^2 : \langle u_{l+1},x\rangle=b_{l+1}\}\) is a vertex of \(P(U,b)\). Here the indices are always calculated \(\bmod\, m\). Moreover, \(v_l\) is a vertex of \(P(U,b)\) if and only if \[\langle u_i, v_l\rangle < b_i \text{ for } i\in\{1,\dots,m\}\setminus\{l,l+1\}.\] As the coordinates of \(v_l\) depends linearly on \(b_l, b_{l+1}\) the inequalities above for \(l=1,\dots,m\) describe the interior of \(A(u)\). Hence we have found a representation \(A(U)=\{b\in\mathbb{R}^m : B\,b\geq 0\}\) for some \(B\in \mathbb{R}^{m(m-2)\times m}\).

For \(b\in\mathop{\mathrm{int}}A(U)\) the volume of the facets \(f_i(b)\), i.e., the length of the edges were already calculated by Stancu [22] and we have \[\begin{align} f_{i}(b) = \mathrm{vol}\,_1(F_i(b)) = &- b_i \left( \frac{\langle u_i, u_{i+1} \rangle}{\sqrt{1 - (\langle u_i, u_{i+1} \rangle)^2}} + \frac{\langle u_{i-1}, u_{i} \rangle}{\sqrt{1 - (\langle u_{i-1}, u_{i} \rangle)^2}} \right) \\ & + \frac{b_{i+1}}{\sqrt{1 - (\langle u_i , u_{i+1} \rangle)^2}} + \frac{b_{i-1}}{\sqrt{1 - (\langle u_{i-1} , u_{i} \rangle)^2}}. \end{align}\] Along with pyramid formula \[\mathrm{vol}\,(P(U,b))=\sum_{i = 1}^m f_{i}(b) \cdot \frac{b_i}{2}.\] we have obtained a representation of the set \(W(U)\).

Although it is easy to get this description \(W(U)\), the computation of \(C_{\tt cv}(U)=\Pi(\mathop{\mathrm{cl}}(W(U)))\) remains challenging, as we must eliminate as many quantifiers as there are columns in \(U\). For instance, if we consider a \(U\), the software Mathematica [37] was unable to generate a quantifier-free output for the left set. Even for quadrilaterals the explicit descriptions of \(C_{\tt cv}(U)\) obtained by Liu, Lu, Sun and Xiong [23] are quite involved.

5 On the non-uniquness of cone-volume vectors↩︎

In this section, we briefly study for \(U \in \mathcal{U}(n,m)\) and \(\gamma \in C_{{\tt cv}}(U) \cap \mathbb{R}_{>0}^m\) the set \[S(U, \gamma) = \{ b \in \mathbb{R}_{\geq 0}^m : \gamma(U, b) = \gamma \},\] consisting of all right hand sides \(b \in \mathbb{R}_{\geq 0}^m\) yielding the same cone-volume vector \(b\). In the case of the simplex, i.e., \(m=n+1\), the cardinality of this set is clearly \(1\) whereas in our example of the parallelepiped it is infinity. Hence, to study its size we use the dimension \(\dim^*(S(U,\gamma))\), where \(\dim^*(S(U,\gamma))\) denotes the dimension of \(S(U,\gamma)\) as a semialgebraic set, [33]. In particular, the cardinality of \(S(U,\gamma)\) is finite if and only if \(\dim^*(S(U,\gamma))=0\). The next proposition gives a lower bound on \(\dim^*(S(U,\gamma))\).

We conjecture that equality holds in the proposition above.

The conjecture suggests that the set \(S(U, \gamma)\) is finite if and only if \(U\) is irreducible. The last example shows that in general it is necessary to assume that the cone-volume vector is strictly positive.

Acknowledgement↩︎

This work is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689).

References↩︎