Gruber P. Convex and Discrete Geometry

Proof. Clearly,
R(p) = �
{i, j}∈E
R{ij} where R{ij} = � x : f{ij}(x) = f{ij}(p) � .
17 Rigidity 299
Since grad f{ij}(p) �= o, the implicit function theorem shows that, in a neighbourhood
of p in Ed ,thesetR{ij} is a smooth hypersurface in Edv with normal vector
grad f{ij}(p) at p. Since, by assumption, the e vectors grad f{ij}(p) are linearly independent,
in a suitable neighbourhood of p in Ed ,thesetR(p) is a smooth manifold
of dimension dv − e. See, e.g. Auslander-MacKenzie [43], p. 32. Since, in this
neighbourhood, C(p) is a sub-manifold of R(p), wehave 1 2d(d + 1) ≤ dv − e or
e ≤ dv − 1 2d(d + 1).
If e < dv − 1 2 (d + 1) then C(p) is a proper sub-manifold of R(p) and we can
choose a continuous function x : [0, 1] → Edv such that (1) holds, i.e. F(p) is
flexible. If e = dv − 1 2 (d + 1) then C(p) has the same dimension as R(p) and thus
coincides with R(p) in a neighbourhood of p. In this case there is no such continuous
function x(t) which satisfies (1). Hence F(p) is rigid. ⊓⊔
Rigidity of Convex Frameworks in E 3
The rigidity predictor theorem and arguments from the proof of Cauchy’s rigidity
theorem yield the following result of Asimow and Roth [41].
Theorem 17.3. Let F(p) be the framework consisting of the vertices and edges of a
proper convex polytope P in E 3 . Then the following statements are equivalent:
(i) All facets of P are triangles.
(ii) F(p) is rigid.
The implication (i)⇒(ii) is an immediate consequence of Cauchy’s rigidity theorem.
Proof. Let p = (p1,...,pv) ∈ E 3v where p1,...,pv are the vertices of P. Let
E ={{i, j} :[pi, p j] is an edge of P} and define f{ij}(x) =�xi − x j � 2 for x =
(x1,...,xv) ∈ E 3v and {i, j} ∈E. Leta(i) ={j :{i, j} ∈E}, i = 1,...,v.
The main step of the proof is to show that
(2) The vectors grad f{ij}(p), {i, j} ∈E, are linearly independent.
To see this, assume the contrary. Then there are real numbers ω{ij}, not all 0, such
that
(3) �
ω{ij} grad f{ij}(p) = o.
Note that
{i, j}∈E
grad f{ij}(p) = 2(o,...o, pi − p j, o,...,o, p j − pi, o,...,o) ∈ E 3v ,
for {i, j} ∈E, or, equivalently, for i ∈{1,...,v} and j ∈ a(i).