Classification of Vertex-Transitive Zonotopes

We give a full classification of vertex-transitive zonotopes. We prove that a vertex-transitive zonotope is a $\Gamma$-permutahedron for some finite reflection group $\Gamma\subset\mathrm O(\Bbb R^d)$. The same holds true for zonotopes in which all vertices are on a common sphere, and all edges are of the same length (which we call homogeneous zonotopes). The classification of these then follows from the classification of finite reflection groups. We provide two alternative characterizations of root systems.


Introduction
A (convex) polytope is the convex hull of finitely many points. This paper deals with a special class of polytopes, the zonotopes, for which several equivalent definitions are known [1]: Definition 1.1. A zonotope is a polytope Z ⊂ R d which satisfies any of the following equivalent conditions: (i) Z is a parallel projection of a δ-cube for some δ ≥ d.
(ii) Z is a Minkowski sum of line segments.
(iii) Z has only centrally symmetric faces (iv) Z has only centrally symmetric 2-faces.
zonopes. We were able to show, that these are all the vertex-transitive zonotopes that exist. Surprisingly (but in fact easier to prove), the same holds for zonotopes for which all vertices are on a common sphere and all edges are of the same length (no symmetry requirements are necessary). These will be called homogeneous zonotopes. Besides a classification, the proof of the above claims also provides interesting properties and alternative characterizations for root systems.
1.1. Overview. Section 2 and Section 3 recap all relevant definitions and preliminary results that are known from the literature. In Section 2 we define the generators of a zonotope (i.e., a canonical way to write the zonotope as a Minkowski sum of line segments) and recall how these determine its faces. We also show that the generators determine the symmetries of the zonotope. Section 3 recalls premutahedra, reflection groups and root systems. We show that permutahedra are exactly the zonotopes generated by root systems. This is the core property exploited in the following sections.
In Section 4 we provide a proof of our main result (Theorem 4.11): the vertextransitive zonotopes are exactly the Γ-permutahedra, where Γ ⊂ O(R d ) is some finite reflection group. We also show, that a homogeneous zonotope (all vertices on a common sphere, all edges of the same length) is necessarily a Γ-permutahedron. The latter is a purely geometric result, as no symmetry requirements are made. These results enable a complete classification of vertex-transitive/homogeneous zonotopes, which we elaborate in detail in Section 4.4. In Section 5 we apply the classification of the previous sections to give alternative characterizations of root systems (Corollary 5.1 and Theorem 5.2). For example, a centrally symmetric set R ⊂ R d can be intersected with a half space, so that the resulting intersection contains exactly half the vectors of R. We show, that if all those resulting "half-sets" are congruent, then R is a root system, and this property actually defines root systems. We also prove a second sufficient condition.

Generators, Faces and Symmetries
Throughout this paper, Z ⊂ R d with d ≥ 2 shall always denote a full-dimensional zonotope. The cases d ∈ {0, 1} are trivial and will not be considered here. Zonotopes are centrally symmetric, and we will assume Z = −Z.
2.1. Generators. By Definition 1.1 (ii), Z is the Minkwoski sum of line segments, and so there is a finite centrally symmetric set We use the convention Zon(∅) := {0}. Conversely, Gen(Z) := {r ∈ R d | conv{−r, r} is the translate of an edge of Z} is a finite, centrally symmetric set, with Zon(Gen(Z)) = Z. The elements of Gen(Z) are called generators of Z, and Z is generated by Gen(Z).
Let R ⊂ R d \ {0} be any finite centrally symmetric set.
The set Gen(Z) of generators is always reduced. If the set R was already reduced, then it is centrally symmetric, and R = Gen(Zon(R)). In other words, R is the set of generators of some zonotope.

2.2.
Faces. For each generator r ∈ Gen(Z), the zonotope has an edge with direction r, and vice versa, each generator is parallel to some edge. This generalizes as follows: If σ is the set of maximizers of ·, c in Z for c ∈ R d , then and these sets are independent of the exact choice of c.
Essentially, Proposition 2.2 states that σ is a certain translate of the zonotope Zon(R 0 ). For example, if σ is a vertex of Z, then necessarily R 0 = ∅. If σ is an edge of Z, then R 0 = {−r, r} for some r ∈ Gen(Z).
Proof of Proposition 2.2. By (2.1), each point v ∈ Z can be written in the form v = r∈Gen(Z) a r r, with coefficients a r ∈ [0, 1].
Fix some arbitrary partition Gen(Z) = R − · ∪ R 0 · ∪ R + . Consider the following two separate sets of constraints for the coefficients a r of v: for r, c < 0 1 for r, c > 0 arbitrary for r, c = 0 The constraints in ( * ) are equivalent to v maximizing the functional ·, c in Z. On the other hand, the constraints in ( * * ) are equivalent to v ∈ r∈R+ r + r∈R0 a r r a ∈ [0, 1] R0 = r∈R+ r + Zon(R 0 ).
By definition, the face σ is the set of all points in Z that satisfy ( * ). If σ is of the form (2.2), then σ is additionally the set of all points in Z that satisfy ( * * ). This happens exactly when ( * ) and ( * * ) are equivalent conditions. Comparing the definitions of ( * ) and ( * * ), wee see that we necessarily need to choose R − , R 0 and R + as claimed.
A subset R ⊆ Gen(Z) is linearly closed, if span(R) ∩ Gen(Z) = R. Evidently, all linearly closed subsets of reduced sets are reduced. For example, the subset R 0 in Proposition 2.2 is linearly closed, hence reduced.
The faces of a zonotope are in a many-to-one correspondence with the linearly closed subsets of Gen(Z): Corollary 2.3. The following holds: (i) For each face σ ∈ F(Z), Gen(σ) ⊆ Gen(Z) is a linearly closed subset.
(ii) For each linearly closed subset R ⊆ Gen(Z) there is at least one σ ∈ F(Z) with Gen(σ) = R.
For (ii), choose a vector c ∈ R d , so that the hyperplane c ⊥ contains the vectors in R, but no vectors in Gen(Z)\R. This is possible, since R is a linearly closed subset. The set σ of maximizers of ·, c in Z is a face of Z. We apply Proposition 2.2 to σ, and observe that the set R 0 (as defined in Proposition 2.2) contains exactly the vectors in R, and no more. Then Gen(σ) = R 0 = R follows via (2.3).
A finite centrally symmetric set R ⊂ R d \ {0} is sometimes also called a star. We shall not adopt this terminology, but it motivated the term "semi-star" in the following definition.
where c ∈ R d satisfies r, c = 0 for all r ∈ R.
A semi-star contains half the elements of a star, namely, the ones in one halfspace of R d . More precisely, it contains one element from each pair {−r, r} ⊆ R. In the case R = Gen(Z), we write Gen c (Z) for a semi-star.
Corollary 2.5. The vertices F 0 (Z) are in one-to-one correspondence with the semistars of Gen(Z). That is, the following map is bijective: Then R 0 = ∅, and R + (as defined in Proposition 2.2) is a semi-star.
2.3. Symmetries. Zon(·) and Gen(·) are linear constructions, and we can directly check that for any orthogonal transformation T ∈ O(R d ) holds
Proof. If T ∈ Aut(Z) is a symmetry of Z, then T Gen(Z) = Gen(T Z) = Gen(Z), and thus, T ∈ Aut(Gen(Z)). The proof of the other direction goes equivalently.
For later use we prove Proposition 2.7. Let R ⊂ R d be a finite centrally symmetric set, and T ∈ O(R d ).
(i) If T ∈ Aut(R), then T maps semi-stars onto semi-stars: For any vector r ∈ R, either r ∈ R c , and then T r ∈ R by assumption, or −r ∈ R c , then T r = −T (−r) ∈ −R = R by centrally symmetry. Thus T R = R and T ∈ Aut(R).

Permutahedra and root systems
The standard (d − 1)-dimensional permutahedron is obtained as the convex hull of all coordinate permutations of the vector (1, 2, ..., d) ∈ R d . These polytopes, and certain generalizations for general reflection groups (the Γ-permutahedra), provide well-known examples for vertex-transitive zonotopes.
3.1. Γ-permutahedra and reflection groups. In the following, Γ ⊂ O(R d ) shall denote a finite reflection group, that is, a matrix group where R ⊂ R d is some finite set of vectors, and T r ∈ O(R d ) denotes the reflection on the hyperplanes r ⊥ . Let E denotes the union of the reflection hyperplanes r ⊥ , r ∈ R.
The connected components of R d \E are called Weyl chambers of Γ. It is well-known that Γ acts regularly (i.e., transitively and freely) on these chambers [6].
Definition 3.1. A Γ-permutahedron is a polytope P ⊂ R d that satisfies any of the following equivalent conditions: (i) Γ acts regularly on the vertices of P .
Remark 3.2. A Γ-permutahedron has exactly one vertex per Weyl chamber of Γ, and no vertices in E (such a vertex would be fixed by a reflection).
A Γ-permutahedron is always vertex-transitive, but not necessarily a zonotope (some faces might not be centrally symmetric). Still, there is always a point v ∈ R d , so that conv(Γv) is a zonotope (see, e.g. the construction in Remark 3.5). Among these points, there is also always one, so that the zonotope has all edge of the same length. The resulting polytopes are classically known as the omnitruncated uniform polytopes [5].
The finite reflection groups are completely classified [6]. The irreducible ones are denoted by I 1 , , the sub-index always denotes the dimension). The reducible ones are obtained as direct sums of the irreducible ones. See Figure 1 for the Γ-permutahedra generated from A 3 (resp. D 3 ), B 3 and H 3 . 3.2. Root systems. Root systems bridge between zonotopes and reflection groups. We shall work with a minimalistic version of the definition (no crystallographic or reducedness restrictions): A zonotope is a Γ-permutahedron if and only if Gen(Z) is a root system (we provide a proof for completeness in Lemma 3.4 and Lemma 3.6).
The Weyl group of a root system R is the group This is a finite reflection group. Note that Γ ⊆ Aut(R) by definition of root system. Lemma 3.4. If Gen(Z) is a root system, then Z is a Γ-permutahedron, where Γ is the Weyl group of Gen(Z).
Proof. It holds Γ ⊆ Aut(Gen(Z)) by definition of root system, and therefore Γ ⊆ Aut(Z) by Proposition 2.6. The proof proceeds in two steps: first, we show that Γ acts transitively on F 0 (Z). This shows that F 0 (Z) is an orbit of Γ. Second, we show that no T r , r ∈ Gen(Z) fixes a vertex of Z. The vertices must then lie in the interior of the Weyl chambers of Γ, and Γ must act regularly on them.
Step 1 : The edge graph of Z is connected. So, for any two vertices v, w ∈ F 0 (Z) the edge graph contains a path v 0 v 1 · · · v k between v = v 0 and w = v k . That means, The reflection T ri fixes e i and exchanges its end vertices v i−1 and v i . The map T := T r k · · · T r1 ∈ Γ now satisfies T v = w. This proves vertex-transitivity.
Step 2 : For v ∈ F 0 (Z), there is a unique semi-star Gen c (Z) whose elements sum up to v (see Corollary 2.5). For any generator r ∈ Gen(Z) exactly one element of {−r, r} is in Gen c (Z). Since T r (±r) = ∓r, the reflection T r cannot fix Gen c (Z). And since the semi-star is unique for v, it cannot fix v either.
To see this, choose a reduced root system R that generates Γ (e.g. build R from the normals of the mirror hyperplanes of Γ). Then, Zon(R) is a zonotope, and a Γ-permutahedron by Gen(Zon(R)) = R and Lemma 3.4. Lemma 3.6. If Z is a Γ-permutahedron, then Gen(Z) is a root system. Proof. Choose a generator r ∈ Gen(Z) and let e ∈ F 1 (Z) be an r-parallel edge. There is at most one vertex per Weyl chamber of Γ (Remark 3.2), hence the end vertices of e cannot be in the same chamber. Therefore, e must cross one of the reflection hyperplanes of Γ. These are symmetry hyperplanes of Z, and if e crosses one, it must be perpendicular to it. Thus, this hyperplane is r ⊥ .
We have shown that T r ∈ Aut(Z) = Aut(Gen(Z)) for all r ∈ R, and therefore, Gen(Z) is a root system.

Vertex-transitive and homogeneous zonotopes
The goal of this section is to prove Theorem 4.11, that all vertex-transitive (and homogeneous, Definition 4.4) zonotopes are Γ-permutahedra.

4.1.
Roadmap. We briefly describe the roadmap to the proof. Section 4.2 discusses 2-dimensional zonotopes, i.e., we argue that 2-dimensional vertex-transitive zonotopes are Γ-permutahedra (where Γ ∈ {I 2 (p), I 1 ⊕ I 2 } is a 2dimensional reflection group). This case is quite trivial, but it serves as a basis for the general result: we can now prove that a zonotope is already a Γ-permutahedron, if all its 2-faces are vertex-transitive (Theorem 4.2).
It is well-known, that the faces of vertex-transitive polytopes are not necessarily vertex-transitive. We therefore cannot apply Theorem 4.2 directly to our vertextransitive zonopes (their faces are vertex-transitive, but we cannot prove it at this point). For this reason, we prove Proposition 4.3, which provides an elementary sufficient condition for a polygon to be vertex-transitive.
In Section 4.3 we define homogeneous zonotopes (a purely geometric construction, without any symmetry requirements), and prove that these are Γ-permutahedra (Corollary 4.6). The proof is surprisingly simple. To transfer the result to vertex-transitive zonotopes, we define the normalization of a zonotope (Definition 4.7), which transforms any vertex-transitive zonotope into a homogeneous one. This transformation does not change much about the shapes of the 2-faces of the zonotope. Therefore, Proposition 4.3 can be used to show, that the 2-faces of the initial vertex-transitive zonotope must have been vertex-transitive. Applying Proposition 4.3 then proves the main result (Theorem 4.11).

2-dimensional zonotopes.
A 2-dimensional zonotope is a centrally symmetric 2n-gon. Such one is vertex-transitive, if either (i) it is a regular 2n-gon, or (ii) n is even, and it has alternating edge lengths, as seen in Figure 2. This list is complete: every vertex-transitive polygons is an orbit polytope to a cyclic group or dihedral group. These orbit polytopes are contained in the list above.
If Z is a 2n-gon as listed in (i) or (ii), Gen(Z) consists of of 2n vectors in R 2 , equally spaced by an angle of π/n, and in the case (ii), these vectors alternate in length (see Figure 2). These are exactly the root systems that corresponds to the reflection group I 2 (n) (if n ≥ 3), or I 1 ⊕ I 1 (if n = 2).
Applying Lemma 3.4, we obtain the classification in dimension two.

Corollary 4.1. A 2-dimensional vertex-transitive zonotope is a Γ-permutahedron.
This already finishes the case of 2-dimensional vertex-transitive zonotopes. As it turns out, investigating vertex-transitive zonotopes in general dimensions comes down to the study of their 2-faces: Proof. Choose generators r, s ∈ Gen(Z). We show T r s ∈ Gen(Z), establishing that Gen(Z) is a root system and Z is a Γ-permutahedron by Lemma 3.4.
The case r = ±s is trivial. We therefore assume that R := Gen(Z) ∩ span{r, s} is 2-dimensional. In particular, R is a linearly closed subset. Corollary 2.3 then gives us the existence of a 2-face σ with Gen(σ) = R. By assumption, σ is vertextransitive, and R therefore a root system (Corollary 4.1). Conclusively, T r s ∈ R ⊆ Gen(Z).
In order to apply Theorem 4.2, we need to prove that certain 2-faces are vertextransitive. This does not follows immediately (even though it is true) from the fact that we start from a vertex-transitive zonotope. Instead, we use the following helper result: (i) has all vertices on a common circle, and (ii) has the same edge directions as a regular 2n-gon, This statement is elementary. We sketch its proof: Proof of Proposition 4.3. A (convex) polygon has at most two edges of the same direction, and if it is centrally symmetric, it has exactly two of each. A regular 2n-gon has n edge directions, and by (ii), so has Z. Since Z is centrally symmetric, it must be a 2n-gon as well.
Let α i ∈ R be the exterior angle of the i-th vertex of Z (see Figure 3). By (ii) we have α i = k i π/n, where k i ∈ N is an integer ≥ 1. The exterior angles of a (convex) polygon add up to 2π, and so we estimate and conclude that k i = 1. That is, all exterior and interior angles are equal. For simplicity, assume that Z, and all poygons mentioned in the following paragraph are of circumradius one. Let be the length of the shortest edge of Z. Then, is no longer than the edge length of a regular 2n-gon. This ensures that there exists a vertex-transitive 2n-gon with an edge of length (consider an appropriately chosen orbit polytope of I 2 (n) resp. I 1 ⊕ I 1 ). But a polygon satisfying (i) and with prescribed identical interior angles at every vertex is already uniquely determined by placing a single edge (the placement of both incident edges follows uniquely from the set restrictions, and this iteratively determines the whole polygon). Therefore, Z must be this vertex-transitive polygon.
4.3. The general case. We now introduce homogeneous zonotopes and show that these must be Γ-permutahedra. We then use this result to proof the main theorem Theorem 4.11. Definition 4.4. A zonotope is said to be homogeneous, if all its vertices are on a common sphere, and all its edges are of the same length.
Homogeneity is a hereditary property: Observation 4.5. The faces of a homogeneous zonotope Z are homogeneous: all edges of a face σ ∈ F(Z) are of the same length. All vertices of Z are on a sphere S, and all vertices of σ are on an affine subspace of R d . All vertices of σ are on the intersection of this subspace with S, which is itself a sphere.
The 2-faces of homogeneous zonotopes are homogeneous, and homogeneous 2faces are regular, thus vertex-transitive. With Theorem 4.2 we conclude Corollary 4.6. If Z is homogeneous, then Z is a Γ-permutahedron.
To apply these results to vertex-transitive zonotopes, we need the following construction: Definition 4.7. The normalization of Z is the zonotope Z * := Zon r r r ∈ Gen(Z) .
The normalized zonotope has the same edge directions as Z, but all edges are of the same length.
We further need to understand how vertex-transitivity is determined by Gen(Z): is a 2-dimensional and linearly closed subset of the root system Gen(Z * ). As such, it is a root system itself. A 2-dimensional root system consists of vectors that are equally spaces by an angle π/n for some n ∈ N. Stated differently, the elements of R * are the edge directions of a regular 2n-gon. The 2-face σ has the edge direction (but not necessarily the edge lengths) from R * , hence, the same edge direction as a regular 2n-gon. Additionally, as a face of a vertex-transitive polytope, σ has all vertices on a common sphere. By Proposition 4.3, σ is vertex-transitive. We found that all 2-faces of Z are vertex-transitive. Theorem 4.2 provides that Z is a Γ-permutahedron.
We summarize the results in the following theorem: Theorem 4.12. If Z is a zonotope, then the following are equivalent: (i) Z is vertex-transitive.
(ii) All semi-stars of Gen(Z) are congruent.
We list some consequences of this classification, none of which holds for general polytopes: Corollary 4.13. The following holds: (i) The faces of a vertex-transitive zonotope are vertex-transitive.
(ii) A homogeneous zonotope (i.e., it has all vertices on a common sphere, and all edges of the same length) is vertex-transitive. (iii) A zonotope in which all faces (in fact, all 2-faces) are homogeneous (resp. vertex-transitive) is itself homogeneous (resp. vertex-transitive).
Proof. If Z is vertex-transitive, then Gen(Z) is a root system (Theorem 4.11). By Corollary 2.3, for every face σ ∈ F(Z), its generators Gen(σ) ⊆ Gen(Z) form a linearly closed subset of the root system Gen(Z), and hence, form a root system as well. Hence, σ is a Γ-permutahedron and vertex-transitive. This proves (i). Part (ii) follows immediately from Corollary 4.6. Part (iii) follows from Theorem 4.2 (a homogeneous 2-face is regular, hence also vertex-transitive).

4.4.
The classification of vertex-transitive/homogeneous zonotopes. We apply the results in Corollary 4.6 and Theorem 4.11 to compile a list of all vertextransitive zonotopes.
We restrict the enumeration to the irreducible vertex-transitive/homogeneous zonotopes, i.e., those which result from irreducible reflection groups. The remaining zonotopes (the reducible ones) are obtained as cartesian products of the irreducible ones.
We obtain the following list of homogeneous zonotopes: (i) infinitely many 2-dimensional homogeneous zonotopes (the regular 2n-gons), (ii) for each d ≥ 3, the d-dimensional zonotopes generated by reflection groups A d , B d and D d (which are distinct if and only if d > 3), and (iii) six exceptional zonotopes to the reflection groups H 3 , H 4 , F 4 , E 6 , E 7 and E 8 in their respective dimensions d ∈ {3, 4, 6, 7, 8}. All of these are uniquely determined up to scale and orientation. The polytopes in that list are also classically known as the omnitruncated uniform polytopes. This terminology, and many related names were coined by Norman Johnson [5].
The list of (irreducible) vertex-transitive zonotopes differs from above list in two points: (i) the 4n-gons form a continuous 1-dimensional family of combinatorially equivalent vertex-transitive zonotopes, and (ii) the zonotopes of the B d -family form a continuous 1-dimensional family of combinatorially equivalent vertex-transitive zonotopes. These two cases are special for the following reason: among the root systems, only I 2 (2n) (or I 1 ⊕ I 1 if n = 1; see Figure 2) and B d consist of two orbits of vectors under their symmetry group (all other root systems have a single orbit). For each orbit, the length of its vectors can be chosen independently from the other orbit, giving each such zonotope one degree of freedom that manifests in two (possibly) different edge lengths.
All reducible vertex-transitive zonotopes have degrees of freedom. For example, the d-cubes (the cartesian product of line segments) belong to the continuous family of d-orthotopes with d degrees of freedom.
We give further information on some of the families: 4.4.1. The family A d . The generated zonotope is the standard permutahedron. For d = 3, this zonotope is called truncated octahedron (Figure 1), which coincides with the zonotope obtained from D 3 (this is the only pair of coinciding zonotopes). The classical name for this family is omnitruncated d-simplices.

4.4.2.
The family B d . The vertices of a general B d -permutahedron are formed by the coordinate permutations and sign selections of some vector If the x i form a linear sequence x i = x 0 + (i − 1) for some fixed x 0 , > 0, then the corresponding polytope is a zonotope. The quotient x 0 / parametrizes the 1dimensional family. For x 0 / = 1/2, we obtain the homogeneous zonotope of that class ( Figure 1). Clasically, the homogeneous zonotopes of this family are called omnitruncated d-cubes.

Characterizing root systems
Our results on vertex-transitive and homogeneous zonotopes enable us to give interesting alternative characterizations of root systems: is finite and reduced, then the following are equivalent: (i) R is a root system, and (ii) all semi-stars of R are congruent.
Proof. R is reduced, hence the set of generator of Zon(R). We can apply (ii) ⇔ (iii) from Theorem 4.12.
The statement of Corollary 5.1 still holds true, even if R is not reduced, but we leave that to the reader.
The norm of a semi-star shall be the norm of the sum of its elements. It follows from Corollary 5.1, that a root system has all semi-stars of the same norm. We prove a weaker form of a converse: Theorem 5.2. Let R ⊂ S d be a finite centrally symmetric set of unit vectors. If all semi-stars of R have the same norm, then R is a root system. A zonotope that has all vertices on a common sphere, but is not a Γ-permutahedron, but is combinatorially equivalent to one. The corresponding generators have all semi-stars of the same norm, but form no root system.

Proof.
Recall that each vertex of Zon(R) can be written as the sum of the vectors in some semi-star of R (Corollary 2.5). The norm of that semi-star therefore equals the distance of that vertex from the origin.
Since all semi-star have the same norm, Zon(R) has all vertices on a common sphere around the origin. Furthermore, R is centrally symmetric, and all vectors in R are of the same length, i.e., all edges of Zon(R) are of the same length. Thus, Zon(R) is homogeneous, and by Corollary 4.6 it is a Γ-permutahedron.
Since R is centrally symmetric and consists of unit vectors, it is reduced. Conclusively, R = Gen(Zon(R)) is a set of generators of a Γ-permutahedron, and therefore a root system.
It is unclear to the author whether central symmetry must be an explicit assumption in Theorem 5.2, or whether it follows from all semi-stars having the same norm (see Question 6.2). The condition R ⊂ S d however is necessary (see Figure 4)

Open questions
Instead of vertex-transitive zonotopes, we can ask for the larger class of zonotopes with all vertices on a common sphere. If we set no restrictions on the edges (as e.g. in the homogeneous case), we obtain a class with much unclearer properties. One reasonable question might be the following: Question 6.1. If a zonotope has all vertices on a common sphere, is it combinatorially equivalent to a Γ-permutahedron?
The answer is trivially positive in two dimensions. See Figure 4 for a zonotope in which all vertices are on a common sphere, but which is not a Γ-permutahedron. The answer is not clear in ≥ 3 dimensions.
Another question, for which no immediate answer was found, is whether it is necessary to assume central symmetry in Theorem 5.2 or whether this property already follows from the other assumptions: Question 6.2. Let S ⊂ S d be a finite set of unit vectors, in which all semi-stars have the same norm. Is S centrally symmetric, hence a root system?