Motivation Preliminaries Rigidity in infinite frameworks Euclidean and polyhedral normed spaces Questions? The Rigidity of Infinite Frameworks in Euclidean and Polyhedral Normed Spaces Sean Dewar Lancaster University, Department of Mathematics June 8, 2017
Motivation Preliminaries Rigidity in infinite frameworks Euclidean and polyhedral normed spaces Questions? Normed spaces A real normed space is a vector space X over R together with a map � · � : X → [0 , ∞ ) such that for all x , y ∈ X and λ ∈ R : � x � = 0 ⇔ x = 0 � λ x � = | λ |� x � � x + y � ≤ � x � + � y � .
Motivation Preliminaries Rigidity in infinite frameworks Euclidean and polyhedral normed spaces Questions? The Euclidean norm � · � 2 on R d is given by � a 2 1 + . . . + a 2 � ( a 1 , . . . , a d ) � 2 = d .
Motivation Preliminaries Rigidity in infinite frameworks Euclidean and polyhedral normed spaces Questions? The Euclidean norm � · � 2 on R d is given by � a 2 1 + . . . + a 2 � ( a 1 , . . . , a d ) � 2 = d . For a centrally symmetric polytope P ⊆ R d with with facets ± F 1 , . . . , ± F n we can define the norm � · � P on R d by � �� � ˆ � x � P = max F k , x � � � � 1 ≤ k ≤ n F ∈ R d is the unique vector that defines the hyperspace that the where ˆ face F lies on.
Motivation Preliminaries Rigidity in infinite frameworks Euclidean and polyhedral normed spaces Questions? Asimow-Roth for normed spaces The following is a famous result from The Rigidity of Graphs by L.Asimow and B. Roth and an equivalent result for polyhedral normed spaces from Finite and Infinitesimal Rigidity with Polyhedral Norms by Derek Kitson. Theorem Let ( G , p ) be a finite, affinely spanning and regular framework in ( R d , � · � 2 ) or ( R d , � · � P ) . Then TFAE: ( G , p ) is infinitesimally rigid ( G , p ) is continuously rigid (all deformations are rigid motions) ( G , p ) is locally rigid (all equivalent frameworks within a neighbourhood of p are congruent). What would be an equivalent result for infinite frameworks in either space?
Motivation Preliminaries Rigidity in infinite frameworks Euclidean and polyhedral normed spaces Questions? Figure: Infinitesimally rigid but continuously flexible in ( R 2 , � · � 2 ). This framework is infinitesimally flexible for all generic positions.
Motivation Preliminaries Rigidity in infinite frameworks Euclidean and polyhedral normed spaces Questions? Frameworks We shall always assume that ( X , � · � ) is a finite dimensional real normed space with an open set of smooth points. Definition A framework in ( X , � · � ) is a pair ( G , p ) where G is a simple graph (i.e. no loops, repeated edges and undirected) and p ∈ X V ( G ) .
Motivation Preliminaries Rigidity in infinite frameworks Euclidean and polyhedral normed spaces Questions? Frameworks We shall always assume that ( X , � · � ) is a finite dimensional real normed space with an open set of smooth points. Definition A framework in ( X , � · � ) is a pair ( G , p ) where G is a simple graph (i.e. no loops, repeated edges and undirected) and p ∈ X V ( G ) . For a framework we will define the rigidity map to be f G : X V ( G ) → R E ( G ) , ( x v ) v ∈ V ( G ) �→ ( � x v − x w � ) vw ∈ E ( G ) .
Motivation Preliminaries Rigidity in infinite frameworks Euclidean and polyhedral normed spaces Questions? Frameworks We shall always assume that ( X , � · � ) is a finite dimensional real normed space with an open set of smooth points. Definition A framework in ( X , � · � ) is a pair ( G , p ) where G is a simple graph (i.e. no loops, repeated edges and undirected) and p ∈ X V ( G ) . For a framework we will define the rigidity map to be f G : X V ( G ) → R E ( G ) , ( x v ) v ∈ V ( G ) �→ ( � x v − x w � ) vw ∈ E ( G ) . Definition We say an edge vw ∈ E ( G ) of ( G , p ) is well-positioned if p v − p w is a smooth point and we say ( G , p ) is well-positioned if all edges ( G , p ) are well-positioned.
Motivation Preliminaries Rigidity in infinite frameworks Euclidean and polyhedral normed spaces Questions? Support functionals For a well-positioned edge vw ∈ E ( G ) we define the linear functional ϕ v , w : X → R to be the support functional of p v − p w .
Motivation Preliminaries Rigidity in infinite frameworks Euclidean and polyhedral normed spaces Questions? Support functionals For a well-positioned edge vw ∈ E ( G ) we define the linear functional ϕ v , w : X → R to be the support functional of p v − p w . For ( R d , � · � 2 ): � p v − p w � ϕ v , w ( · ) = � p v − p w � , · .
Motivation Preliminaries Rigidity in infinite frameworks Euclidean and polyhedral normed spaces Questions? Support functionals For a well-positioned edge vw ∈ E ( G ) we define the linear functional ϕ v , w : X → R to be the support functional of p v − p w . For ( R d , � · � 2 ): � p v − p w � ϕ v , w ( · ) = � p v − p w � , · . For polyhedral normed space ( R d , � · � P ): � � ˆ ϕ v , w ( · ) = F , · � � ˆ where � p v − p w � P = F , p v − p w .
Motivation Preliminaries Rigidity in infinite frameworks Euclidean and polyhedral normed spaces Questions? Notation The space of infinitesimal flexes: u ∈ X V ( G ) : ϕ v , w ( u v − u w ) = 0 for all vw ∈ E ( G ) � � F ( G , p ) =
Motivation Preliminaries Rigidity in infinite frameworks Euclidean and polyhedral normed spaces Questions? Notation The space of infinitesimal flexes: u ∈ X V ( G ) : ϕ v , w ( u v − u w ) = 0 for all vw ∈ E ( G ) � � F ( G , p ) = The space of trivial flexes: � p v (0)) v ∈ V ( G ) ∈ X V ( G ) : γ is a smooth rigid body motion � ( γ ′ T ( p ) =
Motivation Preliminaries Rigidity in infinite frameworks Euclidean and polyhedral normed spaces Questions? Notation The space of infinitesimal flexes: u ∈ X V ( G ) : ϕ v , w ( u v − u w ) = 0 for all vw ∈ E ( G ) � � F ( G , p ) = The space of trivial flexes: � p v (0)) v ∈ V ( G ) ∈ X V ( G ) : γ is a smooth rigid body motion � ( γ ′ T ( p ) = Orbit of p : � � O p := ( h ( p v )) v ∈ V ( G ) : h is an isometry of ( X , � · � )
Motivation Preliminaries Rigidity in infinite frameworks Euclidean and polyhedral normed spaces Questions? Equicontinuity Let F be a family of continuous curves f : I → X for some interval I and some normed space X . We say that F is equicontinuous at t 0 ∈ I if for all ǫ > 0 there exists δ > 0 such that t ∈ ( − δ + t 0 , δ + t 0 ) ⇒ � f ( t 0 ) − f ( t ) � < ǫ for all f ∈ F . If F is equicontinuous at all t ∈ I then F is equicontinuous .
Motivation Preliminaries Rigidity in infinite frameworks Euclidean and polyhedral normed spaces Questions? Equicontinuity Let F be a family of continuous curves f : I → X for some interval I and some normed space X . We say that F is equicontinuous at t 0 ∈ I if for all ǫ > 0 there exists δ > 0 such that t ∈ ( − δ + t 0 , δ + t 0 ) ⇒ � f ( t 0 ) − f ( t ) � < ǫ for all f ∈ F . If F is equicontinuous at all t ∈ I then F is equicontinuous . Definition We say that a family α = ( α v ) v ∈ V ( G ) of continuous paths α v : ( − 1 , 1) → X is an equicontinuous finite flex of ( G , p ) in ( X , � · � ) if: α v (0) = p v for all v ∈ V ( G ) � α v ( t ) − α w ( t ) � = � p v − p w � for all vw ∈ E ( G ) and t ∈ ( − 1 , 1) α is equicontinuous.
Motivation Preliminaries Rigidity in infinite frameworks Euclidean and polyhedral normed spaces Questions? Topology of X V ( G ) For X V ( G ) we define the generalised metric (i.e. a metric that allows infinite distances between points) d V ( G ) where d V ( G ) ( x , y ) := sup � x v − y v � . v ∈ V ( G )
Motivation Preliminaries Rigidity in infinite frameworks Euclidean and polyhedral normed spaces Questions? Topology of X V ( G ) For X V ( G ) we define the generalised metric (i.e. a metric that allows infinite distances between points) d V ( G ) where d V ( G ) ( x , y ) := sup � x v − y v � . v ∈ V ( G ) We now define for all p ∈ X V ( G ) and r > 0 the open balls of X V ( G ) � q ∈ X V ( G ) : d V ( G ) ( p , q ) < r � B r ( p ) := .
Motivation Preliminaries Rigidity in infinite frameworks Euclidean and polyhedral normed spaces Questions? Topology of X V ( G ) For X V ( G ) we define the generalised metric (i.e. a metric that allows infinite distances between points) d V ( G ) where d V ( G ) ( x , y ) := sup � x v − y v � . v ∈ V ( G ) We now define for all p ∈ X V ( G ) and r > 0 the open balls of X V ( G ) � q ∈ X V ( G ) : d V ( G ) ( p , q ) < r � B r ( p ) := . For more information on generalised metric spaces see A Course in Metric Geometry by Dmitri Burago, Yuri Burago and Sergei Ivanov.
Motivation Preliminaries Rigidity in infinite frameworks Euclidean and polyhedral normed spaces Questions? Rigidity for infinite frameworks Definition A framework ( G , p ) is infinitesimally rigid if F ( G , p ) = T ( p ).
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