Finite and infinitesimal rigidity with polyhedral norms

We characterise finite and infinitesimal rigidity for bar-joint frameworks in R^d with respect to polyhedral norms (i.e. norms with closed unit ball P a convex d-dimensional polytope). Infinitesimal and continuous rigidity are shown to be equivalent for finite frameworks in R^d which are well-positioned with respect to P. An edge-labelling determined by the facets of the unit ball and placement of the framework is used to characterise infinitesimal rigidity in R^d in terms of monochrome spanning trees. An analogue of Laman's theorem is obtained for all polyhedral norms on R^2.


Introduction
A bar-joint framework in R d is a pair (G, p) consisting of a simple undirected graph G = (V (G), E(G)) (i.e. no loops or multiple edges) and a placement p : V (G) → R d of the vertices such that p v and p w are distinct whenever vw is an edge of G. Given a norm on R d we are interested in determining when a given framework can be continuously and nontrivially deformed without altering the lengths of the bars. A well-developed rigidity theory exists in the Euclidean setting for finite bar-joint frameworks (and their variants) which stems from classical results of A. Cauchy [6], J. C. Maxwell [16], A. D. Alexandrov [3] and G. Laman [13]. Of particular relevance is Laman's landmark characterisation for generic minimally infinitesimally rigid finite bar-joint frameworks in the Euclidean plane. Asimow and Roth proved the equivalence of finite and infinitesimal rigidity for regular bar-joint frameworks in two key papers [1], [2]. A modern treatment can be found in works of Graver, Servatius and Servatius [8] and Whiteley [23], [24]. More recently, significant progress has been made in topics such as global rigidity ( [5], [7], [10]) and the rigidity of periodic frameworks ( [4], [15], [19], [20]) in addition to newly emerging themes such as symmetric frameworks [21] and frameworks supported on surfaces [17]. In this article we consider rigidity properties of both finite and infinite bar-joint frameworks (G, p) in R d with respect to polyhedral norms. A norm on R d is polyhedral (or a block norm) if the closed unit ball {x ∈ R d : x ≤ 1} is the convex hull of a finite set of points. Such norms are important from a number of perspectives. Firstly, every norm on R d may be approximated by a polyhedral norm. Secondly, polyhedral norms are used in diverse areas of mathematical modelling. Thirdly, the rigidity theory obtained with polyhedral norms is distinctly different to the Euclidean setting in admitting edge-labelling and spanning tree methods. A study of rigidity with respect to the classical non-Euclidean p norms was initiated in [11] for finite bar-joint frameworks and further developed for infinite bar-joint frameworks in [12]. Among these norms the 1 and ∞ norms are simple examples of polyhedral norms and so the results obtained here extend some of the results of [11].
In Section 1 we provide the relevant background material on polyhedral norms and finite and infinitesimal rigidity. In Section 2 we establish the role of support functionals in determining the space of infinitesimal flexes of a bar-joint framework (Theorem 2.5). We then distinguish between general bar-joint frameworks and those which are well-positioned with respect to the unit ball. The well-positioned placements of a finite graph are open and dense in the set of all placements and we show that finite and infinitesimal rigidity are equivalent for these bar-joint frameworks (Theorem 2.7). We then introduce the rigidity matrix for a general finite bar-joint framework, the non-zero entries of which are derived from extreme points of the polar set of the unit ball. In Section 3 we apply an edge-labelling to G which is induced by the placement of each bar in R d relative to the facets of the unit ball. With this edge-labelling we identify necessary conditions for infinitesimal rigidity and obtain a sufficient condition for a subframework to be relatively infinitesimally rigid (Proposition 3.5). We then characterise the infinitesimally rigid bar-joint frameworks with d induced framework colours as those which contain monochrome spanning trees of each framework colour (Theorem 3.9). This result holds for both finite and infinite bar-joint frameworks and does not require the framework to be well-positioned. For minimal infinitesimal rigidity we must assume that the bar-joint framework is wellpositioned and an example is provided to demonstrate this. In Section 4 we apply the spanning tree characterisation to show that certain graph moves preserve minimal infinitesimal rigidity for any polyhedral norm on R 2 . We then show that in two dimensions a finite graph has a well-positioned minimally infinitesimally rigid placement if and only if it satisfies the counting conditions |E(G)| = 2|V (G)|−2 and |E(H)| ≤ 2|V (H)|−2 for all subgraphs H (Theorem 4.9). This is an analogue of Laman's theorem [13] which characterises the finite graphs with minimally infinitesimally rigid generic placements in the Euclidean plane as those which satisfy the counting conditions |E(G)| = 2|V (G)| − 3 and |E(H)| ≤ 2|V (H)| − 3 for subgraphs H with at least two vertices. Many of the results obtained hold equally well for both finite and infinite bar-joint frameworks. We conclude in Section 5 with a discussion of some aspects which are unique to the infinite case. Illustrative examples are provided throughout.

Preliminaries
Let P be a convex symmetric d-dimensional polytope in R d where d ≥ 2. Following [9] we say that a proper face of P is a subset of the form P ∩ H where H is a supporting hyperplane for P. A facet of P is a proper face which is maximal with respect to inclusion. The set of extreme points (vertices) of P is denote ext(P). The polar set of P is denoted P and is also a convex symmetric d-dimensional polytope in R d , Moreover, there exists a bijective map which assigns to each facet F of P a unique extreme pointF of P such that The polar set of P is P.
The Minkowski functional (or gauge) for P defines a norm on R d , This is what is known as a polyhedral norm or a block norm. The dual norm of · P is also a polyhedral norm and is determined by the polar set P , In general, a linear functional on a convex polytope will achieve its maximum value at some extreme point of the polytope and so the polyhedral norm · P is characterised by, A point x ∈ R d belongs to the conical hull cone(F ) of a facet F if x = n j=1 λ j x j for some non-negative scalars λ j and some finite set of points x 1 , x 2 . . . , x n ∈ F . By formulas (1), (2) and (3) the following equivalence holds, Each isometry of the normed space (R d , · P ) is affine (by the Mazur-Ulam theorem) and hence is a composition of a linear isometry and a translation. A linear isometry must leave invariant the finite set of extreme points of P and is completely determined by its action on any d linearly independent extreme points. Thus there exist only finitely many linear isometries on (R d , · P ).
A continuous rigid motion of (R d , · P ) is a family of continuous paths, with the property that α x (0) = x and for every pair x, y ∈ R d the distance α x (t) − α y (t) P remains constant for all values of t. If δ is sufficiently small then the isometries Γ t : x → α x (t) are necessarily translational since by continuity the linear part must equal the identity transformation. Thus we may assume that a continuous rigid motion of (R d , · P ) is a family of continuous paths of the form x ∈ R d for some continuous function c : (−δ, δ) → R d (cf. [12,Lemma 6.2]). An infinitesimal rigid motion of (R d , · P ) is a vector field on R d which arises from the velocity vectors of a continuous rigid motion. Since the continuous rigid motions are initially of translational type, the infinitesimal rigid motions of (R d , · P ) are precisely the constant maps Let (G, p) be a bar-joint framework in (R d , · P ). A continuous (or finite) flex of (G, p) is a family of continuous paths for all |t| < δ and each edge vw ∈ E(G). A continuous flex of (G, p) is regarded as trivial if it arises as the restriction of a continuous rigid motion of (R d , · P ) to p(V (G)). If every continuous flex of (G, p) is trivial then we say that (G, p) is continuously rigid.
An infinitesimal flex of (G, p) is a map u : for each edge vw ∈ E(G). We will denote the collection of infinitesimal flexes of (G, p) by F(G, p). An infinitesimal flex of (G, p) is regarded as trivial if it arises as the restriction of an infinitesimal rigid motion of (R d , · P ) to p(V (G)). In other words, an infinitesimal flex of (G, p) is trivial if and only if it is constant. A bar-joint framework is infinitesimally rigid if every infinitesimal flex of (G, p) is trivial. Regarding F(G, p) as a real vector space with component-wise addition and scalar multiplication, the trivial infinitesimal flexes of (G, p) form a d-dimensional subspace T (G, p) of F(G, p). The infinitesimal flex dimension of (G, p) is the vector space dimension of the quotient space F(G, p)/T (G, p).

Support functionals and rigidity
In this section we begin by highlighting the connection between the infinitesimal flex condition (5) for a general norm on R d and support functionals on (R d , · ). We then characterise the space of infinitesimal flexes for a general bar-joint framework in (R d , · P ) in terms of support functionals and prove the equivalence of finite and infinitesimal rigidity for finite wellpositioned bar-joint frameworks in (R d , · P ). Following this we describe the rigidity matrix for general finite bar-joint frameworks in (R d , · P ) and compute some examples.
2.1. Support functionals. Let · be an arbitrary norm on R d and denote by B the closed unit ball in (R d , · ). A linear functional f : Proof. Since f is linear and f (x 0 ) = x 0 2 we have for all y ∈ R d , Let (G, p) be a bar-joint framework in (R d , · ) and fix an orientation for each edge vw ∈ E(G). We denote by supp(vw) the set of all support functionals for p v − p w . (The choice of orientation on the edges of G is for convenience only and has no bearing on the results that follow. Alternatively, we could avoid choosing an orientation by defining supp(vw) to be the set of all linear functionals which are support functionals for either Proof. Let vw ∈ E(G) and suppose f is a support functional for p v − p w . Applying Lemma 2.1 with x 0 = p v − p w and y = u v − u w we have, Since u is an infinitesimal flex of (G, p), lim t→0 1 t ( x 0 + ty − x 0 ) = 0 and so f (y) = 0.
Let · P be a polyhedral norm on R d . For each facet F of P denote by ϕ F the linear functional 3. Let · P be a polyhedral norm on R d , let F be a facet of P and let x 0 ∈ R d . Then x 0 ∈ cone(F ) if and only if the linear functional, is a support functional for x 0 .
Proof. If x 0 ∈ cone(F ) then by formula (4), ϕ F,x 0 (x 0 ) = x 0 2 P . By (1) we have ϕ F,x 0 (x) ≤ x 0 P for each x ∈ P and it follows that ϕ F,x 0 is a support functional for x 0 . Conversely, if x 0 / ∈ cone(F ) then by (4), ϕ F,x 0 (x 0 ) < x 0 2 P and so ϕ F,x 0 is not a support functional for x 0 .
For each oriented edge vw ∈ E(G) we denote by supp Φ (vw) the set of all linear functionals ϕ F which are support functionals for pv−pw pv−pw P .
ker ϕ F for each edge vw ∈ E(G) then there exists δ > 0 such that the family is a finite flex of (G, p).
Proof. Let vw ∈ E(G) and write x 0 = p v − p w and u 0 = u v − u w . If ϕ F is a support functional for x 0 x 0 P then, by the hypothesis, ϕ F (u 0 ) = 0. By Lemma 2.3, x 0 is contained in the conical hull of the facet F . Applying formulas (3) and (4), x 0 · y = x 0 ·F By continuity there exists δ vw > 0 such that for all |t| < δ vw , Since G is a finite graph the result holds with δ = min vw∈E(G) δ vw > 0.
The following is a characterisation of the space of infinitesimal flexes of a general bar-joint framework in (R d , · P ). Theorem 2.5. Let (G, p) be a bar-joint framework in (R d , · P ). Then a mapping u : V (G) → R d is an infinitesimal flex of (G, p) if and only if Proof. If u is an infinitesimal flex of (G, p) then the result follows from Proposition 2.2. For the converse, let vw ∈ E(G) and write x 0 = p v − p w and u 0 = u v − u w . Applying the argument in the proof of Proposition 2.4, there exists δ vw > 0 with x 0 + tu 0 P = x 0 P for all |t| < δ vw . Hence u is an infinitesimal flex of (G, p).

Equivalence of finite and infinitesimal rigidity.
is contained in the conical hull of exactly one facet of the unit ball P for each edge vw ∈ E(G). We denote this unique facet by F vw . In the following discussion G is a finite graph and each placement is identified with a point p = (p v ) v∈V (G) in the product space v∈V (G) R d which we regard as having the usual topology. The set of all well-positioned placements of G in (R d , · P ) is an open and dense subset of this product space. The configuration space for a bar-joint framework (G, p) is defined as, is a well-positioned bar-joint framework for each x ∈ U , and, Proof. Let vw ∈ E(G) be an oriented edge and consider the continuous map, is an open neighbourhood of p. Since G is a finite graph the intersection, is an open neighbourhood of p which satisfies (i), (ii) and (iii). Since (G, p) is well-positioned, by Lemma 2.3, there is exactly one support functional in supp Φ (vw) for each edge vw and this functional is given by for each edge vw ∈ E(G). By Theorem 2.5, the latter identity is equivalent to the condition that u is an infinitesimal flex of (G, p). Thus x ∈ V (G, p)∩U if and only if x ∈ U and x − p ∈ F(G, p).
We now prove the equivalence of continuous rigidity and infinitesimal rigidity for finite well-positioned bar-joint frameworks.
Theorem 2.7. Let (G, p) be a finite well-positioned bar-joint framework in (R d , · P ). Then the following statements are equivalent.
is an infinitesimal flex of (G, p) then by Theorem 2.5 and Proposition 2.4, the family is a finite flex of (G, p) for some > 0. Since (G, p) is continuously rigid this finite flex must be trivial. Thus there exists δ > 0 and a continuous path and so u is a constant, and hence trivial, infinitesimal flex of (G, p). We conclude that (G, p) is infinitesimally rigid.
(ii) ⇒ (i). If (G, p) has a finite flex given by the family, then consider the continuous path, is a trivial finite flex of (G, p). We conclude that (G, p) is continuously rigid.
The non-equivalence of finite and infinitesimal rigidity for general finite bar-joint frameworks in (R d , · P ) is demonstrated in Examples 2.9 and 3.8.

2.3.
The rigidity matrix. We define the rigidity matrix R P (G, p) for a finite bar-joint framework (G, p) in (R d , · P ) as follows: Fix an ordering of the vertices V (G) and edges E(G) and choose an orientation on the edges of G. For each vertex v assign d columns in the rigidity matrix and label these columns p v,1 , . . . , p v,d . For each directed edge vw ∈ E(G) and each facet F with p v − p w ∈ cone(F ) assign a row in the rigidity matrix and label this row by (vw, F ). The entries for the row (vw, F ) are given by Proof. The system of equations in Theorem 2.5 is expressed by the matrix equation The space of trivial infinitesimal flexes of (G, p) has dimension d and so in general we have If F is a facet of P and y 1 , y 2 , . . . , y d ∈ ext(P) are extreme points of P which are contained in F then for each column vector y k we compute [ Moreover, if y 1 , y 2 , . . . , y d are pairwise orthogonal then where · 2 is the Euclidean norm on R d . Example 2.9. Let P be a crosspolytope in R d with 2d many extreme points ext(P) = {±e k : k = 1, . . . , d} where e 1 , e 2 , . . . , e d is the usual basis in R d . Then each facet F contains d pairwise orthogonal extreme points y 1 , y 2 , . . . , y d each of Euclidean norm 1. By (7),F = d j=1 y j and the resulting polyhedral norm is the 1-norm Consider for example the placements of the complete graph K 2 in (R 2 , · 1 ) illustrated in Figure 1. The polytope P is indicated on the left with facets labelled F 1 and F 2 . The extreme points of the polar set P which correspond to these facets areF 1 = e 1 + e 2 = (1, 1) andF 2 = e 1 − e 2 = (1, −1). The first placement is well-positioned with respect to P and the rigidity matrix is, pv,1 pv,2 pw,1 pw,2 (vw,F1) This bar-joint framework has infinitesimal flex dimension 1. The second placement is not well-positioned and the rigidity matrix is, pv,1 pv,2 pw,1 pw,2 (vw,F1) As the rigidity matrix has rank 2 this bar-joint framework is infinitesimally rigid in (R 2 , · 1 ), but continuously flexible.

Edge-labellings and monochrome subgraphs
In this section we describe an edge-labelling on G which depends on the placement of the bar-joint framework (G, p) in (R d , · P ) relative to the facets of P. We provide methods for identifying infinitesimally flexible frameworks and subframeworks which are relatively infinitesimally rigid. We then characterise infinitesimal rigidity for bar-joint frameworks with d framework colours in terms of the monochrome subgraphs induced by this edge-labelling.
3.1. Edge-labellings. Let (G, p) be a general bar-joint framework in (R d , · P ). Since P is symmetric in R d , if F is a facet of P then −F is also a facet of P. Denote by Φ(P) the collection of all pairs We refer to the elements of Φ(vw) as the framework colours of the edge vw. For example, if p v − p w lies in the conical hull of exactly one facet of P then the edge vw has just one framework colour. If p v − p w lies along a ray through an extreme point of P then vw has at least d distinct framework colours. By Lemma 2.3, [F ] is a framework colour for an edge vw if and only if either Proof. If v 0 ∈ V (G) and |Φ(v 0 )| < d then there exists non-zero then u is a non-trivial infinitesimal flex of (G, p).
We now consider the subgraphs of G which are spanned by edges possessing a particular framework colour. For each facet F of P define and let G F be the subgraph of G spanned by E F (G, p). We refer to G F as a monochrome subgraph of G.  (7),F = 1 d ( d j=1 y j ) = ±e k for some k. The resulting polyhedral norm is the maximum norm, For example, consider the placement p of the complete graph K 3 in (R 2 , · ∞ ) illustrated in Figure 2. The polytope P is indicated on the left with facets labelled F 1 and F 2 . This bar-joint framework is well-positioned with respect to P as each edge has exactly one framework colour, The monochrome subgraphs G F1 and G F2 are indicated in black and gray respectively. The corresponding extreme points of P areF 1 = (1, 0) andF 2 = (0, 1). The rigidity matrix has rank 3 and so (K 3 , p) has infinitesimal flex dimension 1.
The edges which are incident with the vertex c each have framework colour [F 2 ] and so a non-trivial infinitesimal flex of (K 3 , p) may be obtained as in the proof of Proposition 3.1.

Φ(vw)
We refer to the elements of Φ(G, p) as the framework colours of the bar-joint framework. (G, p) Proposition 3.3. Let (G, p) be an infinitesimally rigid bar-joint framework in (R d , · P ). If C is a collection of framework colours of (G, p) with |Φ(G, p)\C| < d then contains a spanning tree of G.
Proof. Suppose that [F ]∈C G F does not contain a spanning tree of G. Then there exists a partition then u is a non-trivial infinitesimal flex of (G, p). We conclude that [F ]∈C G F contains a spanning tree of G.
The converse to Proposition 3.3 does not hold in general as the following example illustrates. In Theorem 3.9 we show that a converse statement does hold under the additional assumption that |Φ(G, p)| = d.
Every norm of this type is a polyhedral norm. If F is a facet of the closed unit ball and x is an interior point of the conical hull of F then Consider the polyhedral norm on R 2 given by 0), b 2 = (0, 1) and b 3 = (1, 1). Let (K 3 , p) be the bar-joint framework in (R 2 , · P ) which is illustrated in Figure 3. The monochrome subgraphs corresponding to the facets F 1 , F 2 and F 3 are indicated by black, gray and dashed lines respectively. The rigidity matrix is Note that if C is any collection of at least two framework colours then |Φ(G, p)\C| < 2 and [F ]∈C G F contains a spanning tree of G. However, the rigidity matrix has rank 3 and so the infinitesimal flex dimension of (K 3 , p) is 1.  3.2. Edge-labelled paths and relative infinitesimal rigidity. For each edge vw ∈ E(G) let X vw be the vector subspace of R d , For each pair of vertices v, w ∈ V (G) denote by Γ G (v, w) the set of all paths γ in G from v to w. A subframework of (G, p) is a bar-joint framework (H, p) obtained by restricting p to the vertex set of a subgraph H. We say that (H, p) is relatively infinitesimally rigid in (G, p) if the restriction of every infinitesimal flex of (G, p) to (H, p) is trivial.
Proposition 3.5. Let (G, p) be a bar-joint framework in (R d , · P ) and let (H, p) be a subframework of (G, p). If for each pair of vertices v, w ∈ V (H), then (H, p) is relatively infinitesimally rigid in (G, p).
Proof. Let u ∈ F(G, p) be an infinitesimal flex of (G, p) and let v, w ∈ V (H).
Since this holds for all paths in Γ G (v, w) the hypothesis implies that u v = u w . Applying this argument to every pair of vertices in H we see that the restriction of u to V (H) is constant and hence a trivial infinitesimal flex of (H, p). Thus (H, p) is relatively infinitesimally rigid in (G, p). Example 3.6. Let (G, p) be the bar-joint framework in (R 2 , · ∞ ) indicated in Figure 4 and let H be the subgraph of G induced by the vertices v 1 , v 2 , v 3 . The monochrome subgraphs G F1 and G F2 are indicated in black and gray respectively. Each pair of vertices in H is connected by a path in G F1 and a path in G F2 and so, by Proposition 3.5, (H, p) is relatively infinitesimally rigid in (G, p). Corollary 3.7. Let (G, p) be a bar-joint framework in (R d , · P ) and let u ∈ F(G, p) be an infinitesimal flex. If vw ∈ E(G) and p v − p w lies in a ray passing through an extreme point of P then u v = u w .
Example 3.8. Let (K 1,n , p) be a placement of the bipartite graph K 1,n with edges v 0 v 1 , v 0 v 2 , . . . , v 0 v n in (R d , · P ) such that v 0 is placed at the origin and all other vertices are placed at extreme points of P. This bar-joint framework is not wellpositioned as each edge has at least d distinct framework colours. It follows from Corollary 3.7 that (K 1,n , p) is infinitesimally rigid (but continuously flexible). Consider for example the class of polyhedral norms on R 2 for which P is an n-gon with extreme points v k = cos 2π(k−1) n , sin 2π(k−1) n where n ∈ 2Z, n ≥ 4 and k = 1, 2, . . . , n. Then P has n facets F 1 , F 2 , . . . , F n where F k is the closed line segment from v k to v k+1 . Applying (6), the corresponding extreme point of the polar P isF k = sec π n cos (2k−1)π n , sin (2k−1)π n . The case n = 8 is illustrated in Figure 5 with P an octagon in R 2 . Each edge contributes two independent rows to the rigidity matrix R P (K 1,8 , p). For example the entries for the row v 0 v 1 are, In particular, the rigidity matrix for (K 1,8 , p) has rank 2|E(K 1,8 )| = 2|V (K 1,8 )|− 2 (cf. Proposition 2.8).  Proof. The implication (i) ⇒ (ii) follows from Proposition 3.3. To prove (ii) ⇒ (i) let u ∈ F(G, p). If v, w ∈ V (G) then for each framework colour [F ] ∈ Φ(G, p) there exists a path in G F from v to w. Hence and, by Proposition 3.5, u v = u w . Applying this argument to all pairs v, w ∈ V (G) we see that u is a trivial infinitesimal flex and so (G, p) is infinitesimally rigid.
A bar-joint framework (G, p) is minimally infinitesimally rigid in (R d , · P ) if it is infinitesimally rigid and every subframework obtained by removing a single edge from G is infinitesimally flexible. Corollary 3.10. Let (G, p) be a bar-joint framework in (R d , · P ) and suppose that |Φ(G, p)| = d. If G F is a spanning tree in G for each [F ] ∈ Φ(G, p) then (G, p) is minimally infinitesimally rigid.
Proof. By Theorem 3.9, (G, p) is infinitesimally rigid. If any edge vw is removed from G then G F is no longer a spanning tree for some [F ] ∈ Φ(G, p). By Theorem 3.9, the subframework (G\{vw}, p) is not infinitesimally rigid and so we conclude that (G, p) is minimally infinitesimally rigid.
Example 3.11. Let (G, p) be the bar-joint framework in (R 2 , · ∞ ) which is illustrated in Figure 6. This bar-joint framework is well-positioned with respect to P and the subgraphs G F1 and G F2 are indicated in black and gray respectively. Both monochrome subgraphs are spanning trees of G and so, by Corollary 3.10, (G, p) is minimally infinitesimally rigid.  Figure 6. A minimally infinitesimally rigid bar-joint framework in (R 2 , · ∞ ) The converse statement to Corollary 3.10 which we now prove requires the additional assumption that (G, p) is well-positioned. The necessity of this condition is demonstrated in Example 3.13. Corollary 3.12. Let (G, p) be a well-positioned bar-joint framework in (R d , · P ) and suppose that |Φ(G, p)| = d. Then the following statements are equivalent. p). If (G, p) is minimally infinitesimally rigid then by Theorem 3.9, the monochrome subgraph G F contains a spanning tree of G. Suppose vw is an edge of G which is contained in G F . Since (G, p) is minimally infinitesimally rigid, (G\{vw}, p) is infinitesimally flexible. Since (G, p) is wellpositioned, vw is contained in exactly one monochrome subgraph of G and so G F is the only monochrome subgraph which is altered by removing the edge vw from G. By Theorem 3.9, G F \{vw} does not contain a spanning tree of G. We conclude that G F is a spanning tree of G. The implication (ii) ⇒ (i) is proved in Corollary 3.10.
Example 3.13. Let (G, p) be the bar-joint framework in (R 3 , · ∞ ) which is illustrated in Figure 7. The polytope P is the cube with extreme points ±(1, 1, 1), ±(1, 1, −1), ±(1, −1, 1), ±(−1, 1, 1) and the polyhedral norm is the maximum norm, This bar-joint framework is not well-positioned as each edge has two framework colours, The monochrome subgraphs G F1 , G F2 and G F3 are indicated in blue, red and green respectively and each contains a spanning tree of G. Thus by Theorem 3.9, (G, p) is infinitesimally rigid. The corresponding extreme points of P areF 1 = (1, 0, 0), F 2 = (0, 1, 0) andF 3 = (0, 0, 1). The rigidity matrix R P (G, p) is The rigidity matrix has rank 3|V (G)| − 3 = 9. By removing the edge ad the resulting rigidity matrix has rank 7 and so the subframework (G\{ad}, p) has infinitesimal flex dimension 2. Removing any other edge results in a subframework with infinitesimal flex dimension 1. Hence (G, p) is minimally infinitesimally rigid. However, G F1 is not itself a spanning tree and this demonstrates the necessity in the hypothesis of Corollary 3.12 that (G, p) is well-positioned.  4. An analogue of Laman's theorem In this section we address the problem of whether there exists a combinatorial description of the class of graphs for which a minimally infinitesimally rigid placement exists in (R d , · P ). We restrict our attention to finite bar-joint frameworks and prove that in two dimensions such a characterisation exists (Theorem 4.9). This result is analogous to Laman's theorem [13] for bar-joint frameworks in the Euclidean plane and extends [11,Theorem 4.6] which holds in the case where P is a quadrilateral.

Regular placements.
Let ω(G, R d , P) denote the set of all well-positioned placements of a finite simple graph G in (R d , · P ). A bar-joint framework (G, p) is regular in (R d , · P ) if the function achieves its maximum value at p. A finite simple graph G is (minimally) rigid in (R d , · P ) if there exists a well-positioned placement of G which is (minimally) infinitesimally rigid.
Example 4.2. The complete graph K 4 is minimally rigid in (R 2 , · P ) for every polyhedral norm · P . To see this let F 1 , F 2 , . . . , F n be the facets of P and let x 0 ∈ ext(P) be any extreme point of P. Then x 0 is contained in exactly two facets, F 1 and F 2 say. Choose a point x 1 in the relative interior of F 1 and a point x 2 in the relative interior of F 2 . Then by formulas (3) and (4), Since (x 0 ·F 1 ) = (x 0 ·F 2 ) = x 0 P = 1, if x 1 and x 2 are chosen to lie in a sufficiently small neighbourhood of x 0 then by continuity we may assume, We may also assume without loss of generality that Define a placement p : V (K 4 ) → R 2 by setting To determine the framework colours for the remaining edges we will apply the above identities together with formulas (3) and (4).
Consider the edge v 2 v 3 . If k = 1 and is sufficiently small then applying (8), Also by (8) and (11) we have, We conclude that F 1 is the unique facet of P for which p v3 −p v2 P = (p v3 −p v2 )·F 1 and so Consider the edge v 0 v 3 . Applying (9) and (10), for k = 1, 2 we have, By applying (12), and by (9), Finally, consider the edge v 1 v 2 . Applying (12) we have, and this value is positive provided is sufficiently small. By (8) we have, By making a small perturbation we can assume that p v2 − p v1 is contained in the conical hull of exactly one facet of P and so (G, p) is well-positioned. This framework colouring is illustrated in Figure 8 with monochrome subgraphs G F1 and G F2 indicated in black and gray respectively and G F k indicated by the dotted line.
Suppose u ∈ F(K 4 , p). To show that u is a trivial infinitesimal flex we apply the method of Proposition 3.5. The vertices v 0 and v 1 are joined by monochrome paths in both G F1 and G F2 and so u v0 = u v1 . The vertices v 2 and v 3 are also joined by monochrome paths in both G F1 and G F2 and so u v2 = u v3 . The vertices v 1 and v 2 are joined by monochrome paths in G F2 and G F k and so u v1 = u v2 . Thus u is a constant and hence trivial infinitesimal flex of (K 4 , p). We conclude that (K 4 , p), and all regular and well-positioned placements of K 4 , are infinitesimally rigid. In Euclidean space it is often necessary to use a stronger notion of genericity for bar-joint frameworks which requires that all subframeworks of (K V (G) , p) be regular frameworks. Here K V (G) is the complete graph on the vertices of G. In the Euclidean setting (and more generally for the classical p norms with p ∈ (1, ∞)), such placements form an open and dense subset of v∈V (G) R d (see for example [12,Lemma 2.7]). The following example shows that in the case of polyhedral norms such placements need not exist. Example 4.3. Consider a well-positioned placement p of the complete graph K 6 in (R 2 , · ∞ ). The induced framework colouring of the edges of K 6 contains a monochrome subgraph G F which itself contains a copy of the complete graph K 3 . The subframework (K 3 , p) has infinitesimal flex dimension 2. Since the regular placements of K 3 have infinitesimal flex dimension 1, (K 3 , p) is not regular. Thus there does not exist a well-positioned placement of K 6 in (R 2 , · ∞ ) for which all subframeworks are regular. More generally, it follows from Ramsey's theorem that given any polyhedral norm on R d there exists a complete graph for which no such well-positioned placements exists.

4.2.
Counting conditions. The Maxwell counting conditions [16] state that a finite minimally infinitesimally rigid bar-joint framework (G, p) in Euclidean space for all subgraphs H. The following analogous statement holds for polyhedral norms. Proof. If (G, p) is minimally infinitesimally rigid then by Proposition 2.8 the rigidity matrix R P (G, p) is independent and, The rigidity matrix for any subframework of (G, p) is also independent and so A graph G is (d, d)-tight if it satisfies the counting conditions in the above proposition. The class of (2, 2)-tight graphs has the property that every member can be constructed from a single vertex by applying a sequence of finitely many allowable graph moves (see [17,18]). The allowable graph moves are: (1) The Henneberg 1-move (also called vertex addition, or 0-extension).
Proof. Suppose (G, p) is well-positioned and infinitesimally rigid and let G → G be a Henneberg 1-move on the vertices v 1 , v 2 ∈ V (G). Choose distinct [F 1 ], [F 2 ] ∈ Φ(P) and define a placement p of G by p v = p v for all v ∈ V (G) and Then (G , p ) is well-positioned and the edges v 0 v 1 and v 0 v 2 have framework colours [F 1 ] and [F 2 ] respectively. If u ∈ F(G , p ) then the restriction of u to V (G) is an infinitesimal flex of (G, p). This restriction must be trivial and hence constant. In particular, u v1 = u v2 . By Theorem 2.5, ϕ F1 (u v0 − u v1 ) = 0 and ϕ F2 (u v0 − u v1 ) = ϕ F2 (u v0 − u v2 ) = 0 and so u v0 = u v1 . We conclude that (G , p ) is infinitesimally rigid.
A Henneberg 2-move G → G removes an edge v 1 v 2 from G and adjoins a vertex v 0 together with three edges v 0 v 1 , v 0 v 2 and v 0 v 3 .
Proof. Suppose (G, p) is well-positioned and infinitesimally rigid and let G → G be a Henneberg 2-move on the vertices v 1 , v 2 , v 3 ∈ V (G) and the edge v 1 v 2 ∈ E(G).
Hence the restriction of u to V (G) is an infinitesimal flex of (G, p) and must be trivial. In particular, u v1 = u v3 . Now ϕ F1 (u v0 − u v1 ) = 0 and ϕ F2 (u v0 − u v1 ) = ϕ F2 (u v0 − u v3 ) = 0 and so u v0 = u v1 . We conclude that u is a constant and hence trivial infinitesimal flex of (G , p ).  Then (G , p ) is well-positioned. Suppose u ∈ F(G , p ) is an infinitesimal flex of (G , p ). The framework colours for the edges v 0 v 1 and v 0 v 2 are [F 2 ] and [F 1 ] respectively. Thus there exists a path from v 0 to v 1 in the monochrome subgraph G F1 given by the edges v 1 v 2 , v 2 v 0 and there exists a path from v 0 to v 1 in the monochrome subgraph G F2 given by the edge v 0 v 1 . By the relative rigidity method of Proposition 3.5, u v0 = u v1 . If an edge v 1 w in G has framework colour [F ] induced by (G, p) and is replaced by v 0 w in G then the framework colour is unchanged. Thus applying Theorem 2.5, and so the restriction of u to V (G) is an infinitesimal flex of (G, p). This restriction is constant since (G, p) is infinitesimally rigid and so u is a trivial infinitesimal flex of (G , p ).
A vertex-to-K 4 move G → G replaces a vertex v 0 ∈ V (G) with a copy of the complete graph K 4 by adjoining three new vertices v 1 , v 2 , v 3 and six Each edge v 0 w of G which is incident with v 0 may be left unchanged or replaced by one of v 1 w, v 2 w or v 3 w.
Proposition 4.8. The vertex-to-K 4 move preserves infinitesimal rigidity for finite well-positioned bar-joint frameworks in (R 2 , · P ).
Proof. Suppose (G, p) is well-positioned and infinitesimally rigid and let G → G be a vertex-to-K 4 move on the vertex v 0 ∈ V (G) which introduces new vertices v 1 , v 2 and v 3 . Since v 0 has finite valence, there exists an open ball B(p v0 , r) such that if p v0 is replaced with any point x ∈ B(p v0 , r) then (G, x) and (G, p) induce the same framework colouring on G. Let (K 4 ,p) be the well-positioned and infinitesimally rigid placement of K 4 constructed in Example 4.2. Define a well-positioned placement p of G by setting p v = p v for all v ∈ V (G) and where > 0 is chosen to be sufficiently small so that p v1 , p v2 and p v3 are each contained in B(p v0 , r). Suppose u ∈ F(G , p ). By the argument in Example 4.2, the restriction of u to the vertices v 0 , v 1 , v 2 , v 3 is constant. Thus if v 0 w is an edge of G with framework colour [F ] which is replaced by v k w in G then applying Theorem 2.5, and so the restriction of u to V (G) is an infinitesimal flex of (G, p). Since (G, p) is infinitesimally rigid this restriction is constant and we conclude that u is a trivial infinitesimal flex of (G , p ).
We now show that the class of finite graphs which have minimally infinitesimally rigid well-positioned placements in (R 2 , · P ) is precisely the class of (2, 2)-tight graphs. In particular, the existence of such a placement does not depend on the choice of polyhedral norm on R 2 . Theorem 4.9. Let G be a finite simple graph and let · P be a polyhedral norm on R 2 . The following statements are equivalent.
Proof. (i) ⇒ (ii). If G is minimally rigid then there exists a placement p such that (G, p) is minimally infinitesimally rigid in (R 2 , · P ) and the result follows from Proposition 4.4.
(ii) ⇒ (i). If G is (2, 2)-tight then there exists a finite sequence of allowable graph moves, Every placement of K 1 is certainly infinitesimally rigid. By Propositions 4.5-4.8, for each graph in the sequence there exists a well-positioned and infinitesimally rigid placement in (R 2 , · P ). In particular, (G, p) is infinitesimally rigid for some well-positioned placement p. If a single edge is removed from G then by Proposition 4.4, the resulting subframework is infinitesimally flexible. Hence (G, p) is minimally infinitesimally rigid in (R 2 , · P ).
The collection of placements of a (2, 2)-tight graph which are well-positioned and minimally infinitesimally rigid in (R 2 , · P ) varies with P and this is illustrated in the following two examples.
Example 4.10. Let (G, p) be the well-positioned bar-joint framework in (R 2 , · ∞ ) illustrated in Figure 9. The monochrome subgraph G F2 (indicated in gray) is not a spanning subgraph of G and so, by Theorem 3.9, (G, p) is infinitesimally flexible. The graph G is (2, 2)-tight and so, by Theorem 4.9, the regular placements of G are infinitesimally rigid. We conclude that (G, p) is not a regular bar-joint framework.  Figure 9. A non-regular, infinitesimally flexible placement of a (2, 2)-tight graph in (R 2 , · ∞ ) In the following example we consider the same bar-joint framework as in Example 4.10 but with a different polyhedral norm. In this case the placement is infinitesimally rigid.
). If f is submodular, monotone and satisfies f (∅) = 0 and f ({j}) > 0 for each j then the function x P =f (|x 1 |, . . . , |x d |) is a polyhedral norm on R d . If F is a facet of P and x ∈ cone(F ) • then F = (sgn(x 1 ) σ(1) f (|x|), . . . , sgn(x d ) σ(d) f (|x|)) where σ : {1, . . . , d} → {1, . . . , d} is the inverse of the permutation k → j k determined by the coordinates of x. Consider, for example, the submodular function f : 2 S → R where S = {1, 2} and, The associated polyhedral norm is defined for x = (x 1 , x 2 ) ∈ R 2 by, Let (G, p) be the bar-joint framework in (R 2 , · P ) illustrated in Figure 10. The monochrome subgraphs induced by the facets F 1 , F 2 and F 3 are indicated in black, gray and dashed lines respectively and the corresponding extreme points of the polar set P are,F The rigidity matrix R P (G, p) is, By computing the rank of the rigidity matrix or alternatively by applying the edge-labelled path argument of Proposition 3.5 we see that (G, p) is minimally infinitesimally rigid.  Figure 10. A minimally infinitesimally rigid bar-joint framework in (R 2 , · P )

Infinite frameworks
In this section we consider some aspects of rigidity which are unique to infinite bar-joint frameworks. Let B(p v , r) be the open ball in R d with centre p v and radius r. An infinite bar-joint framework (G, p) is uniformly well-positioned in (R d , · P ) if there exists r > 0 such that (G, x) is well-positioned for all x ∈ v∈V (G) B(p v , r). An equicontinuous flex of (G, p) is a finite flex {α v : v ∈ V (G)} which is also equicontinuous as a collection of functions from an interval (−δ, δ) into R d . An infinite graph is locally finite if every vertex has finite valence.
Proposition 5.1. Let (G, p) be a uniformly well-positioned bar-joint framework in (R d , · P ) and suppose that G is locally finite. If (G, p) is infinitesimally rigid then every equicontinuous flex of (G, p) is trivial.
Proof. Since (G, p) is uniformly well-positioned there exists r > 0 such that (G, x) is well-positioned for all x ∈ U := v∈V (G) B(p v , r). In particular, x v − x w is contained in the conical hull of the same unique facet F vw as p v − p w for each edge vw ∈ E(G). Hence x v − x w P = (x v − x w ) ·F vw for all x ∈ U and all vw ∈ E(G). As in Proposition 2.6 we have V (G, p) ∩ U = (p + F(G, p)) ∩ U where we now regard U as an open neighbourhood of p with respect to the box topology on v∈V (G) R d . If {α v : v ∈ V (G)} is an equicontinuous flex of (G, p) then there exists δ > 0 such that α v (t) ∈ B(p v , r) for all |t| < δ. Thus (α v (t)) v∈V (G) ∈ V (G, p) ∩ U for all |t| < δ. Now (α v (t) − p v ) v∈V (G) ∈ F(G, p) is an infinitesimal flex of (G, p) for each |t| < δ and so must be trivial. Hence there exists c(t) ∈ R d such and the graph is locally finite. Hence by Proposition 5.1, the equicontinuous finite flexes of (G, p) are necessarily trivial. Figure 11. A countable bar-joint framework which is infinitesimally rigid in (R 2 , · ∞ ) but not sequentially infinitesimally rigid.