Equivalence of continuous, local and infinitesimal rigidity in normed spaces

We present a rigorous study of framework rigidity in general finite dimensional normed spaces from the perspective of Lie group actions on smooth manifolds. As an application, we prove an extension of Asimow and Roth's 1978/9 result establishing the equivalence of local, continuous and infinitesimal rigidity for regular bar-and-joint frameworks in a d-dimensional Euclidean space. Further, we obtain upper bounds for the dimension of the space of trivial motions for a framework and establish the flexibility of small frameworks in general non-Euclidean normed spaces.


Introduction
A framework (G, p) is an embedding p of the vertices of a simple graph G into a given normed space. With a given framework a natural question is whether it is in some sense "rigid". In Euclidean spaces many types of rigidity such as global, redundant and universal have been studied intensely [11] [25] [7]. We wish to detect whether a framework in a general normed space is structurally rigid in the sense that either any continuous motion of the vertices that preserves the edge lengths corresponds to an isometric motion of the embedded vertices (continuous rigidity) or that the embedding is locally unique up to an isometric map (local rigidity). In Euclidean space, one method is to consider the a priori stronger notion of infinitesimal rigidity as this implies local and continuous rigidity (see for example [6]). We shall prove the below theorem (found in Section 4): Theorem 1.1. Let (G, p) be a constant (see Section 4) finite framework in a finite dimensional normed space X, then the following are equivalent: (i) (G, p) is infinitesimally rigid in X, (ii) (G, p) is locally rigid in X, (iii) (G, p) is continuously rigid in X.
In 1978/9 L. Asimow and B. Roth proved Theorem 1.1 in the special case where X is Euclidean [3] [4]. Recently research has been undertaken into framework rigidity in non-Euclidean normed spaces, in particular spaces with ℓ p norms (p ∈ [1, ∞]) [16], polyhedral norms [14] and matrix norms such as the Schatten p-norms [15]. For recent research into infinitesimal rigidity we refer the reader to [24] [12] for frameworks with symmetry, [13] [10] for infinitesimal rigidity concerning alternative types of frameworks and [22] for frameworks on surfaces.
In Section 3 we will present a rigorous study of the orbit and the trivial motion space of a set of points. We will give an upper bound for the dimension of the space of trivial motions which will be achievable by most placements. Utilising this we shall in Section 4 prove Theorem 1.1.
In Section 5 we shall obtain further bounds on the dimension of the space of trivial infinitesimal motions for frameworks that lie on some hyperplane of the normed space. These results will allow us to prove that no small framework (a framework with less vertices than the dimension of the normed space plus one) on two or more vertices is infinitesimally rigid in a non-Euclidean space (see Theorem 5.8).

Preliminaries
All normed spaces (X, · ) shall be assumed to be over R and finite dimensional; further we shall denote a normed space by X when there is no ambiguity. For any normed space X we shall use the notation B r (x), B r [x] and S r [x] for the open ball, closed ball and the sphere with centre x and radius r > 0 respectively. We shall define a normed space to be a Euclidean space if its norm is generated by an inner product, otherwise X is a non-Euclidean (normed) space.
Given normed spaces X, Y we shall denote by L(X, Y ) the normed space of all linear maps from X to Y with the operator norm · op and A(X, Y ) to be space of all affine maps from X to Y with the norm topology. If X = Y we shall abbreviate to L(X) and A(X) and if Y = R with the standard norm we define X * := L(X, R) and refer to the operator norm as · when there is no ambiguity. We denote by ι the identity map on X.
For a C 1 -differentiable manifold M we shall denote by T x M the tangent space of M at x ∈ M and T M := x∈M T x M . For a general reference on the theory of manifolds we refer the reader to [19, Section 3].
is continuous then we say that f is C 1 -differentiable on U ′ and define df to be the Some of the results referenced refer specifically to Gâteaux differentiation, however we will only consider Lipschitz maps between finite dimensional normed spaces and in this case Gâteaux differentiability is equivalent to differentiability by [5,Proposition 4.3]. If we were to observe the connection between the rigidity map and the rigidity operator of an infinite framework this would not hold to be true.
For C k -manifolds M and N with k ∈ N ∪ {0, ∞} we define a map f : If f also has a C k -differentiable inverse then it is a C k -diffeomorphism. For k ≥ 1 we may define for each x ∈ M the maps df (x) : T x M → T f (x) N and df : T M → T N that will be consistent if M, N are normed spaces, we refer the reader to [19,Section 3.3] for more detail.
2.2. Support functionals, smoothness and strict convexity. Let x ∈ X and f ∈ X * , then we say that f is support functional of x if f = x and f (x) = x 2 . By an application of the Hahn-Banach theorem it can be shown that every point must have a support functional. We say that a non-zero point x is smooth if it has a unique support functional and define smooth(X) ⊆ X \ {0} to be the set of smooth points of X. If smooth(X) ∪ {0} = X then we say that X is smooth. We define a norm to be strictly convex if tx + (1 − t)y < 1 for all distinct x, y ∈ S 1 [0] and t ∈ (0, 1).
The dual map of X is the map ϕ : smooth(X) ∪ {0} → X * that sends each smooth point to its unique support functional and ϕ(0) = 0. It is immediate that ϕ is homogeneous since f is the support functional of x if and only if af is the support functional of ax for a = 0.
Remark 2.2. If X is Euclidean with inner product ·, · then all non-zero points are smooth and we have ϕ(x) = x, · where x, · : y → x, y . Proposition 2.3. For any normed space X the following properties hold: The set smooth(X) is dense in X and smooth(X) c has measure zero with respect to the Lebesgue measure on X (iv) The map ϕ is continuous.
Proof. x ϕ(x) is continuous on smooth(X), thus ϕ is continuous on smooth(X) also. As ϕ(x) = x it follows that ϕ is continuous at 0 ∈ X * also as required.
2.3. Isometry groups. We shall define Isom(X, · ) to be the group of isometries of (X, · ) and Isom Lin (X, · ) to be the group of linear isometries of X with the group actions being composition; we shall denote these as Isom(X) and Isom Lin (X) if there is no ambiguity. It can be seen by Mazur-Ulam's theorem [26] that all isometries of a finite dimensional normed space are affine i.e. each isometry is the unique composition of a linear isometry followed by a translation, thus Isom(X) has the topology inherited from A(X). It follows from the Closed Subgroup theorem [19, Theorem 5.1.14] that for any normed space the group of isometries is a Lie group (a smooth finite dimensional manifold with smooth group operations) while the group of linear isometries is a compact Lie group since it is closed and bounded in L(X). Lemma 2.4. Let X be a d-dimensional normed space, then the following holds: (i) There exists a unique Euclidean space (X, · 2 ) such that Isom(X, · ) is a subgroup and a closed smooth submanifold of Isom(X, · 2 ) and Isom Lin (X, · ) is a subgroup and a closed smooth submanifold of Isom Lin (X, · 2 ). (ii) If X is Euclidean then: (iii) If X is non-Euclidean then: 2.4. Placements and bar-joint frameworks. We shall assume that all graphs are simple i.e. no loops or parallel edges, however we will allow them to have a countably infinite vertex set unless we explicitly state otherwise. We will denote V (G) and E(G) to be the vertex and edge sets of G respectively. If H is a subgraph of G we will represent this by H ⊆ G. For a set S we shall denote by K S the complete graph on the set S. Let X be a normed space. For any set S we say p ∈ X S is a placement of S in X; we will denote this (p, S) if we need to clarify what set p is the placement of. For a graph G we say p is a placement of G in X if p is a placement of V (G). We define a (bar-joint) framework to be a pair (G, p) where G is a graph and p is a placement of G in X. For all X and S we will gift X S the product topology from X; if |S| < ∞ we define the norm . . , n} is affinely independent. It is immediate that if p is in general position and |S| ≥ dim X + 1 then p is spanning. We denote the set of placements of S in general position by G(S) ⊆ X S ; likewise for any graph G we let G(G) := G(V (G)). If S is finite then G(S) is an open dense subset of X S and X S \ G(S) has measure zero; we can see this as G(S) is the complement of an algebraic set.
For placements (q, T ), (p, S) we say (q, T ) is a subplacement of (p, S) (or (q, T ) ⊆ (p, S)) if T ⊆ S and p v = q v for all v ∈ T . For frameworks (H, q) and (G, p) we say (H, q) is a subframework of (G, p) (or (H, q) ⊆ (G, p)) if H ⊆ G and p v = q v for all v ∈ V (H). If H is also a spanning subgraph we say that (H, q) is a spanning subframework of (G, p).

Trivial motions of placements
3.1. Structure of the orbit of a placement. Let Γ be a Lie group and M a (finite dimensional) smooth manifold. If there exists a smooth group action we say that φ is a Lie group action of Γ on M . We define the following for all x ∈ M : We say that Γ acts properly on M if the map is proper i.e. the preimage of any compact set is compact. If H is a closed subgroup of Γ then by [19, Theorem 5.1.16] Γ/H (the set of left cosets gH, g ∈ Γ) has a unique manifold structure such that the quotient map π : Γ → Γ/H is a smooth surjective submersion i.e. dπ(g) is surjective for all g ∈ Γ. For any set S, x ∈ X S and affine map g ∈ A(X) we define g.x := (g(x v )) v∈S . With this notation we define for any S the map If |S| < ∞ then this is a Lie group action of Isom(X) on X S ; we shall always refer to this group action if we mention Isom(X) acting on X S . Lemma 3.2. For any X and |S| < ∞ the group of isometries Isom(X) acts properly on X S . Proof. Let ((g n .p n , p n )) n∈N be a convergent sequence in the image of θ : Isom(X) × X S → X S × X S with limit (q, p). By Mazur-Ulam's theorem [26] that for each n ∈ N there exists G n ∈ Isom Lin (X) and x n ∈ X such that g n = T xn • G n , where T xn is the translation map y → y + x n . As ((g n .p n , p n )) n∈N converges then (g n .p n ) n∈N and (p n ) n∈N are bounded in X S , thus (x n ) n∈N is bounded as it follows by Bolzano-Weierstrass that we have a convergent subsequence (x n k ) k∈N with limit x ∈ X. Since Isom Lin (X) is compact, there exists a convergent subsequence (G n k l ) l∈N of (G n k ) k∈N with limit G ∈ Isom Lin (X); this implies (g n k l ) l∈N converges to g := T x • G. As ((g n , p n )) n∈N has a convergent subsequence it follows that θ is proper as required.
Proof. By Lemma 3.2 and Lemma 3.1 it follows that O p is a closed smooth submanifold of X V (G) diffeomorphic to Isom(X)/ Stab p under the diffeomorphismφ p .
Proof. Define the (finite dimensional) linear spaces We note that the linear mapρ : Op is a smooth diffeomorphism.
We will define a continuous path through a placement p in X S to be a family α : Let u ∈ X S . If there exists a trivial finite motion α of p that is differentiable at t = 0 and u v = α ′ v (0) for all v ∈ S then we say that u is a trivial (infinitesimal) motion of p. For any placement p we shall denote T (p) to be the the set all trivial infinitesimal motions of p.
Theorem 3.5. Let p be a placement in X, then O p is a smooth manifold with tangent space T (p) at p andφ Proof. Choose a finite subplacement (q, T ) of (p, S) so that the p and q affinely span the same space, then by Lemma 3.3, O q is a smooth manifold diffeomorphic to Isom(X)/ Stab q under the smooth diffeomorphismφ q . By Lemma 3.4, O p is a smooth manifold diffeomorphic to Isom(X)/ Stab p and the restriction map ρ : Corollary 3.6. The map φ p is a smooth submersion and dφ p (ι) : Proof. By Theorem 3.5,φ p is a smooth diffeomorphism. We note that φ p =φ p • π where π : Isom(X) → Isom(X)/ Stab p is the natural quotient map. By the Closed Subgroup theorem [19, Theorem 5.1.14] π is a smooth submersion, thus φ p is a smooth submersion also and dφ p (ι) is surjective. Asφ p is a smooth diffeomorphism then ker dφ p (ι) = ker π as required.
Proof. (iii): Choose a finite subplacement (r, U ) ⊆ (q, T ) ⊆ (p, S) so that the affine span of Define the restriction maps ρ T : O q → O r and ρ S : O p → O r , then by Lemma 3.4, ρ T , ρ S are smooth diffeomorphisms. As ρ = ρ −1 T • ρ S then it is also a smooth diffeomorphism.
(i): If (q, T ) ⊆ (p, S) this follows from iii. Suppose (q, T ) is not a subplacement of (p, S). Define (t, S ⊔ T ) to be the placement where t| S = p and t| T = q. We note all three placements have the same affine span of their placement points. As (p, S) ⊂ (t, S ⊔T ) and (q, T ) ⊆ (t, S ⊔T ) then by part iii we have O p ∼ = O t ∼ = O q as required.
(ii): By Theorem 3.5, O p has tangent space T (p) at p and O q has tangent space T (q) at q. By part i, O p ∼ = O q and so dim T (p) = dim T (q).

Upper and lower bounds of the dimension of T (p).
Proposition 3.8. The following hold for any placement p: (i) dφ p (ι) is injective if and only if φ p is a smooth local diffeomorphism i.e. dφ p (g) is bijective for all g ∈ Isom(X). (ii) φ p is injective if and only if φ p is a smooth diffeomorphism. If either i or ii hold then dim T (p) = Isom(X).
Proof. If φ p is a local diffeomorphism it follows dφ p (ι) is bijective. Suppose dφ p (ι) is injective. Choose g ∈ Isom(X), then dim ker dφ p (g) = 0; this follows as there exists non-zero u ∈ ker dφ p (g) with corresponding smooth curve α : (−1, 1) → Isom(X) (i.e. α(0) = g, α ′ (0) = u) then the curve t → g −1 α(t) generates a non-zero tangent vector at ι that lies in the kernel of dφ p (ι). By this it follows that dφ p (ι) is injective if and only if dφ p (g) is injective for all g ∈ Isom(X). By Corollary 3.6 it follows that dφ p (g) is bijective for all g ∈ Isom(X) and so is a local diffeomorphism.
If φ p is injective then Stab p is trivial, thus Isom(X)/ Stab p = Isom(X) andφ p = φ p . By Theorem 3.5 it then follows φ p is a smooth diffeomorphism. Conversely suppose φ p is a smooth diffeomorphism, then as φ p =φ p • π it follows that the quotient map π is a diffeomorphism. This implies π is a group isomorphism, thus Stab p is trivial and φ p is injective.
If either i or ii hold then dφ p (ι) is bijective and dim T (p) = Isom(X).
Definition 3.9. We define a placement p to be full if φ p is a local diffeomorphism and isometrically full if φ p is a diffeomorphism.
It is immediate that any isometrically full placement is full. By part i of Proposition 3.8 our notion of full agrees with that given in [15]. The set of full placements of a set S will be denoted by Full(S) and likewise the set of full placements of a graph G will be denoted by Full(G). Proof. Suppose g.p = p and choose v 0 , . . . , v d ∈ S so that p v 0 , . . . , p v d is an affine basis of X, then g(p v i ) = p v i for all i = 0, . . . , d. By Mazum-Ulam's theorem [26] g is affine and so since p v 0 , . . . , p v d is an affine basis of X this map must be unique. As ι.p = p then g = ι and φ p is injective. The result now follows by part ii of Proposition 3.8.
Example 3.11. We shall denote by X the space R 2 with the ℓ 3 -norm. The linear isometries of X are generated by the π/2 anticlockwise rotation around the origin and the reflection in the line {(t, 0) : t ∈ R}. Let S = {v 1 , v 2 } and (p, S) and (q, S) be the non-spanning placements in X where p v 1 = q v 1 = 0, p v 2 = (1, 0) and q v 2 = (1, 2). Both placements are full in X, however q is isometrically full while p is not. This example shows that while all spanning placements are isometrically full and all isometrically full placements are full the reverse is not necessarily true.  Proof. Suppose all isometrically full placements in X are spanning, then it follows by part ii of Proposition 3.8 that for all linear hyperplanes Y of X there exists a linear map T Y = ι that is invariant on Y . By [2, (4.7)] X is Euclidean. Conversely suppose X is Euclidean. If p is a non-spanning placement then p lies in some affine hyperplane H. We note that if h is the reflection in H then h.p = p, thus by part ii of Proposition 3.8 p is not isometrically full.
We may now give an upper and lower bound for the dimension of T (p).
Theorem 3.14. For any placement p in a d-dimensional space X, with dim T (p) = dim Isom(X) if and only if p is full.
A rigid motion of X is a family γ := (γ x ) x∈X of continuous maps γ x : (−δ, δ) → X, x ∈ X (for some fixed δ > 0) where γ x (0) = x and γ x (t) − γ y (t) = x − y for all x, y ∈ X and t ∈ (−δ, δ). The following shows that our definition of a trivial finite motion agrees with the definition given in [16] if a framework is isometrically full.
Proposition 3.15. Let p be an isometrically full placement in X. If α is a continuous path through p in X S then the following are equivalent: (i) α is a trivial finite motion.
(ii) There exists a unique continuous path h : There exists a unique rigid motion γ such that γ pv = α v for all v ∈ S.
Proof. (i ⇒ ii): As α is a continuous path in O p and φ p is a smooth diffeomorphism we define the unique continuous path h := φ −1 p • α. (ii ⇒ iii): Define γ to be the unique family of maps γ where γ x (t) = h t (x) for all x ∈ X and t ∈ (−δ, δ), then γ is a rigid motion as required.
(iii ⇒ i): We note that γ restricted to the set {p v : v ∈ S} is a trivial finite motion.  Proof. By part iii of Proposition 2.3 the set smooth(X) is dense and its compliment has measure zero, thus the result holds for all graphs with a single edge. Suppose the result holds for all graphs with n − 1 edges and let G be any graph with n edges. Choose vw ∈ E(G), and define G 1 , G 2 to be the graphs on V (G) where E(G 1 ) = E(G) \ {vw} and E(G 2 ) = {vw} then W(G 1 ) c and W(G 2 ) c have measure zero by assumption. As W(G) c = W(G 1 ) c ∪ W(G 2 ) c then W(G) c has measure zero also; this further implies W(G) is also dense. The result now follows by induction.

Equivalence of continuous, local and infinitesimal rigidity
We define the rigidity map of G (in X) to be the continuous map and for well-positioned placements p we also define the rigidity operator of G at p in X to be the continuous linear map For any framework we define the configuration space of (G, p) in X to be the set is continuous.
Proof. This follows from part iv of Proposition 2.3.
For a finite graph G we say that a well-positioned framework (G, p) is regular if for all q ∈ W(G) we have rank df G (p) ≥ rank df G (q). We shall denote the subset of W(G) of regular placements of G by R(G).
We define a finite flex of a framework (G, p) to be a continuous path α through a placement p where α v (t) − α w (t) = p v − p w for all vw ∈ E(G) and t ∈ (−δ, δ). If α is a trivial finite motion of a placement p of G we say α is a trivial finite flex of (G, p); we note that α will automatically be a finite flex of (G, p) as isometries preserve distance.
If the only finite flexes of (G, p) are trivial then (G, p) is continuously rigid (in X); (G, p) will be defined to be continuously flexible if it is not continuously rigid. For a finite framework (G, p) we say (G, p) is locally rigid (in X) if there exists a neighbourhood likewise we shall define a framework to be locally flexible if it is not locally rigid. We classify these as types of finite rigidity.
We define u ∈ X V (G) to be a trivial (infinitesimal) flex of (G, p) if u is a trivial motion of p. If (G, p) is well-positioned we say that u ∈ X V (G) is an (infinitesimal) flex of (G, p) if df G (p)u = 0. The following proposition shows a link between finite and infinitesimal flexes for frameworks. Lemma 4.5. Let (G, p) be a well-positioned framework in X and α a finite flex of (G, p) that is differentiable at 0, then (α ′ v (0)) v∈V (G) is an infinitesimal flex of (G, p). Since all trivial flexes of (G, p) are trivial motions of p we shall also denote T (p) to be the set all trivial infinitesimal flexes (G, p). If (G, p) is well-positioned we define F(G, p) to be the space of all infinitesimal flexes of (G, p). The latter is clearly a linear space as it is exactly the kernel of the rigidity operator. By Proposition 4.5 it follows T (p) ⊆ F(G, p).
A well-positioned framework (G, p) is infinitesimally rigid (in X) if every flex is trivial and infinitesimally flexible (in X) otherwise. We shall define a well-positioned (G, p) framework to be independent if the rigidity operator of G at p, df G (p), is surjective (or equivalently, if G is finite, |E(G)| = rank df G (p)) and define (G, p) to be dependent otherwise. If a framework is infinitesimally rigid and independent we shall say that it is isostatic. We shall use the convention that any framework with no edges (regardless of placement) is independent and that (K 1 , p) is isostatic for any choice of placement p. It is immediate that if a framework is independent then its placement is regular, however the reverse does not necessarily hold. Remark 4.6. In the setting of Euclidean space, all of the above definitions agree with those used in [3] [4].
Lemma 4.7. Let (G, p) be a finite (possibly not spanning) framework in a d-dimensional normed space X with |V (G)| ≥ d + 1. Suppose q ∈ R(G) is full, then the following hold: (i) If (G, p) is independent then (G, p) is regular and (G, q) is independent.
(ii) If (G, p) is infinitesimally rigid then (G, p) is regular, p is full and (G, q) is infinitesimally rigid.
Proof. (i): As (G, p) is independent then df G (p) is surjective. As surjective linear maps have maximal possible rank then (G, p) is regular. Since q is regular it follows that (G, q) is independent.
(ii): As (G, q) is regular then by the Rank-Nullity theorem we have thus by Theorem 3.14, dim T (q) ≤ dim T (p) ≤ dim Isom(X). As q is full then by Theorem 3.14, dim T (q) = dim Isom(X). It follows that dim T (p) = dim Isom(X) and thus p is full. From the inequality it also follows that (G, q) is infinitesimally rigid. Proof.
. We now note df H (q)(x| V (H) ) = a as required.
The following gives us some necessary and sufficient conditions for infinitesimal rigidity. The following gives an equivalence for isostaticity.
Proposition 4.10. Let (G, p) be a well-positioned framework in X. If any two of the following properties hold then so does the third (and (G, p) is isostatic): Proof. Apply the Rank-Nullity theorem to the rigidity operator of G at p. The result follows the same method as [8, Lemma 2.6.1.c].
Using the results from the last section we may now give a stronger result for independent frameworks. is an open dense subset of X V (G) and G(G) c has measure zero the set R(G) ∩ G(G) is non-empty, thus we choose p ′ to be a regular placement of G in general position. Since (G, p ′ ) is regular it follows that it is also independent.
Define q := p ′ | V (H) , then (H, q) is in general position. As (H, q) ⊆ (G, p ′ ) then by Proposition 4.8, (H, q) is independent; furthermore as H has at least d + 1 vertices then q is spanning. By Theorem 4.9 we have |E(H)| = (dim X)|V (H)| − dim F(H, q). By Corollary 3.10 and Proposition 3.8, dim T (q) = dim Isom(X), thus as T (q) ⊂ F(H, q) we have the required inequality.

4.2.
Proof of Theorem 1.1. For a finite graph G we say that a well-positioned framework (G, p) is constant if there is a neighbourhood N (p) ⊂ W(G) of p such that rank df G (q) = rank df G (p) for all q ∈ N (p). We shall denote C(G) to be the subset of W(G) of constant placements of G.
For Euclidean spaces R(G) = C(G) as R(G) is an open dense subset of X V (G) (see [3,Section 3] for more details). Remark 4.13. Suppose G is any finite graph and smooth(X) is an open subset of X (an example would be any ℓ d q space). We note W(G) will be an open subset of X V (G) and so by Lemma 4.4, R(G) will be an open subset of X V (G) . It now follows that every regular placement will be constant, thus by Theorem 1.1, if (G, p) is infinitesimally rigid then it will be continuously and locally rigid also.

Flexibility of small frameworks and stronger bounds for T (p)
For any placement p in X we shall define T 2 (p) to denote the space of trivial motions of p in (X, · 2 ), the unique Euclidean space for (X, · ) as defined in Lemma 2.4. If we refer to just X we shall be referring to the general normed space (X, · ).
Lemma 5.1. T (p) is a linear subspace of T 2 (p).
For Euclidean spaces we have the following equality for the dimension of the space of trivial motions for non-spanning placements.
We now wish to obtain an upper and lower bound for the dimension of the space of trivial motions for non-spanning placements. To do this we shall first find an upper-bound for when |S| = 2 in non-Euclidean normed spaces and then use an inductive argument. Proof. Since Isom Lin (X) is compact then Isom Lin (X) gives rise to a proper Lie group action on X by x → T (x) for all T ∈ Isom Lin (X), x ∈ X. As O(x 0 ) is the orbit of x 0 (with respect to Isom Lin (X)) then by Lemma 3.1, O(x 0 ) is a closed smooth submanifold of X.
First suppose X is Euclidean. By [26, Corollary 3.3.3] Isom Lin (X) acts transitively on S x 0 [0], thus O(x 0 ) = S x 0 [0]. As the unit sphere of a Euclidean space is the d-sphere and S  Now suppose dim T (p) = 2d − 1. Let S = {v 1 , v 2 }, then without loss of generality we may assume p v 1 = 0. If d = 1 the result holds trivially so we may suppose d > 1. As p is in general position then p v 2 = 0. By Lemma 5.3, O(p v 2 ) is a closed smooth submanifold of X. It follows from [19,Lemma 3.3.4] that we may identify the tangent space of O(p v 2 ) at p v 2 with a subspace of X; by calculation we see that the tangent space of O(p v 2 ) at p v 2 is {u ∈ X : (0, u) ∈ T (p)}. Since dim T (p) = d + (d − 1) and the trivial motions generated by translations form a ddimensional subspace then it follows O(p v 2 ) has dimension d − 1. By Lemma 5.3 the space X is Euclidean as required.
By part ii of Corollary 3.7, T (q) ∼ = T (p) and T (q ′ ) ∼ = T (p ′ ) and so the result follows.
Proposition 5.6. Let (p, S) be a placement in a d-dimensional normed space X where {p v : v ∈ S} has an affine span of dimension 1 ≤ n ≤ d. Then dim T (p) ≤ (n + 1)(2d − n) 2 with equality if and only if X is Euclidean.
Proof. If X is Euclidean then the result follows by Lemma 5.2.
We define a framework (G, p) in a d-dimensional normed space to be small if |V (G)| ≤ d + 1. The following is a well known result for Euclidean spaces. , thus G is a complete graph. Suppose p is not in general position, then there exists distinct vertices v 0 , . . . , v n , w ∈ V (G) such that the affine span of p v 0 , . . . , p vn is Y ⊂ X, the affine span of {p v : v ∈ V (G)}. Define for each v ∈ V (G) the vector u v where u v = 0 if v = w and u w ∈ (Y − p v 0 ) ⊥ \ {0}, then u := (u v ) v∈V (G) ∈ X V (G) is a infinitesimal flex of (G, p). If u is trivial then by Corollary 3.6 there exists an affine map g ∈ T ι Isom(X) such that g.p = u. As p v 0 , . . . , p vn is an affine basis of Y and g(p v i ) = 0 for all i = 0, 1, . . . , n then g(y) = 0 for all y ∈ Y . However g(p w ) = 0 and so no such affine map g may exist. This implies u is a non-trivial flex which contradicts the infinitesimal rigidity of (G, p). Using Theorem 3.14 we can now state our own result for small frameworks for non-Euclidean spaces.  Remark 5.9 (Final remark). We remark that combinatorial characterisations of infinitesimal rigidity have recently been obtained in some normed space contexts. However other forms of rigidity, such as redundant and global rigidity, have not yet been explored in general normed spaces.