Asymptotic properties of some random polytopes Pierre Calka INRIA - PowerPoint PPT Presentation

Asymptotic properties of some random polytopes Pierre Calka INRIA Sophia Antipolis, 9 December 2010

default Outline Random convex hulls in the unit-ball Random tessellations in R d . Typical cell Convergence of the boundary and consequences Joint work with Tomasz Schreiber (Toru´ n University, Poland) and Joseph Yukich (Lehigh University, USA)

default Outline Random convex hulls in the unit-ball Poisson point process Model Historical background Random tessellations in R d . Typical cell Convergence of the boundary and consequences

default Poisson point process �� �� �� �� � Poisson point process of intensity measure µ � � �� �� � :=locally finite set X of R d such that �� �� �� �� � � � � �� �� B 4 ◮ #( X ∩ B 1 ) Poisson r.v. of mean µ ( B 1 ) B 1 � � � � � ◮ #( X ∩ B 1 ) , · · · , #( X ∩ B n ) independent � �� �� �� �� �� �� ( B 1 , · · · , B n ∈ B ( R d ), B i ∩ B j = ∅ , i � = j ) � � B 2 � � B 3 � � If µ = λ d x , X homogeneous of intensity λ � � � �� �� �

default Random convex hulls in the unit-ball ◮ X homogeneous Poisson point process in the unit-ball B d of intensity λ > 0 (resp. set of n independent and uniform points) ◮ Convex hull K λ (resp. K ′ n ) of X

default Historical background Non-asymptotic results ◮ Sylvester (1864): four-point problem 2 35 3 ≤ p C ≤ 1 − 12 π 2 p C := probability that 4 i.i.d. uniform points in a convex set C ⊂ R 2 are extreme d − 1 � n − 1 � ◮ Wendel (1962): P { 0 �∈ K ′ n } = 2 − ( n − 1) � ( n ≥ d ) k k =0 ◮ Efron (1965): f 0 ( · ):= # vertices, V d ( · ):=volume, κ d := V d ( B d ) κ d E f 0 ( K ′ κ d − E V d ( K ′ � � n ) = n n − 1 ) Buchta (2005): identities relating higher moments

default Historical background LLN R´ enyi & Sulanke (1963), Wieacker (1978), Schneider & Wieacker (1980), Schneider (1988) , B´ any & Buchta (1993) ar´ 2 d − 1 κ d − V d ( K λ ) ≈ c d λ − f 0 ( K λ ) ≈ c ′ d +1 , d λ d +1 CLT Reitzner (2005), Vu (2006), Schreiber & Yukich (2008), B´ ar´ any & Reitzner (2009), B´ ar´ any et al. (2009) Aims Description of the boundary of K λ CLT with variance asymptotics

default Outline Random convex hulls in the unit-ball Random tessellations in R d . Typical cell Poisson-Voronoi tessellations Poisson hyperplane tessellations Zero-cell and typical cell Distributional and asymptotic results Connection with random convex hulls Convergence of the boundary and consequences

default Poisson-Voronoi tessellations ◮ X Poisson point process in R d of intensity measure d x ◮ For a nucleus x ∈ X , cell C ( x | X ) := { y ∈ R d : � y − x � ≤ � y − x ′ � ∀ x ′ ∈ X } ◮ Tessellation: set of cells C ( x | X ) Properties : isotropic, stationary, ergodic. References : Descartes (1644), Gilbert (1961), Okabe et al. (1992)

default Poisson hyperplane tessellations ◮ X Poisson point process in R d of intensity measure � x � α − d d x ( α ≥ 1) ◮ For x ∈ X , polar hyperplane H x := { y ∈ R d : � y − x , x � = 0 } ◮ Tessellation: set of connected components of R d \ ∪ x ∈ X H x Properties : isotropic, stationary iff α = 1, ergodic. References : Meijering (1953), Miles (1964), Stoyan et al. (1987)

default Poisson hyperplane tessellations ◮ X Poisson point process in R d of intensity measure � x � α − d d x ( α ≥ 1) ◮ For x ∈ X , polar hyperplane H x := { y ∈ R d : � y − x , x � = 0 } ◮ Tessellation: set of connected components of R d \ ∪ x ∈ X H x Properties : isotropic, stationary iff α = 1, ergodic. References : Meijering (1953), Miles (1964), Stoyan et al. (1987)

default Poisson hyperplane tessellations ◮ X Poisson point process in R d of intensity measure � x � α − d d x ( α ≥ 1) ◮ For x ∈ X , polar hyperplane H x := { y ∈ R d : � y − x , x � = 0 } ◮ Tessellation: set of connected components of R d \ ∪ x ∈ X H x Properties : isotropic, stationary iff α = 1, ergodic. References : Meijering (1953), Miles (1964), Stoyan et al. (1987)

default Poisson hyperplane tessellations ◮ X Poisson point process in R d of intensity measure � x � α − d d x ( α ≥ 1) ◮ For x ∈ X , polar hyperplane H x := { y ∈ R d : � y − x , x � = 0 } ◮ Tessellation: set of connected components of R d \ ∪ x ∈ X H x Properties : isotropic, stationary iff α = 1, ergodic. References : Meijering (1953), Miles (1964), Stoyan et al. (1987)

default Poisson hyperplane tessellations ◮ X Poisson point process in R d of intensity measure � x � α − d d x ( α ≥ 1) ◮ For x ∈ X , polar hyperplane H x := { y ∈ R d : � y − x , x � = 0 } ◮ Tessellation: set of connected components of R d \ ∪ x ∈ X H x Properties : isotropic, stationary iff α = 1, ergodic. References : Meijering (1953), Miles (1964), Stoyan et al. (1987)

default Poisson hyperplane tessellations ◮ X Poisson point process in R d of intensity measure � x � α − d d x ( α ≥ 1) ◮ For x ∈ X , polar hyperplane H x := { y ∈ R d : � y − x , x � = 0 } ◮ Tessellation: set of connected components of R d \ ∪ x ∈ X H x Properties : isotropic, stationary iff α = 1, ergodic. References : Meijering (1953), Miles (1964), Stoyan et al. (1987)

default Poisson hyperplane tessellations ◮ X Poisson point process in R d of intensity measure � x � α − d d x ( α ≥ 1) ◮ For x ∈ X , polar hyperplane H x := { y ∈ R d : � y − x , x � = 0 } ◮ Tessellation: set of connected components of R d \ ∪ x ∈ X H x Properties : isotropic, stationary iff α = 1, ergodic. References : Meijering (1953), Miles (1964), Stoyan et al. (1987)

default Zero-cell and typical cell ◮ Zero-cell: unique cell containing the origin (Crofton cell for a stationary Poisson hyperplane tessellation) ◮ Typical cell: cell C ’chosen uniformly’ among all cells Three equivalent ways to define its distribution: ◮ limits of ergodic means ( Reference : Cowan (1977)) ◮ use of a Palm measure ( Reference : Mecke (1967)) ◮ explicit realization: Poisson-Voronoi tessellation: D C = C (0 | X ∪ { 0 } ) zero-cell of a hyperplane 0 tessellation ( α = d )

default Distributional and asymptotic results ◮ Explicit integral formula for the distribution of the number of hyperfaces and conditional distribution of the cell ◮ D. G. Kendall ’s conjecture (40s): ’when it is large, the zero-cell is close to the spherical shape’. ( Hug , Reitzner & Schneider , 2004) C 0 Particular case: zero-cell conditioned on having R M large inradius R m r 0 ( R M − r )

default Connection with random convex hulls B d 0 R − 1 m C 0 R m := inradius of the zero-cell C 0 of a hyperplane tessellation Effect of the inversion I ( x ) := x / � x � 2 , x � = 0 on R − 1 m C 0 : I ◮ Point process outside B d → point process in B d − I ◮ Line process → Germ-grain process in B d − ◮ Number of hyperfaces of C 0 = number of extreme points in B d

default Outline Random convex hulls in the unit-ball Random tessellations in R d . Typical cell Convergence of the boundary and consequences Radius-vector function and support function Scaling limits Paraboloid growth process Paraboloid hull process Central limit theorems and variance asymptotics Brownian limits Extreme values Dual results for zero-cells of hyperplane tessellations

default Radius-vector function and support function s λ ( v ) B d r λ ( u ) 0 Defect radius-vector function r λ ( u ) := 1 − sup { r > 0 : ru ∈ K λ } , u ∈ S d − 1 Defect support function s λ ( u ) := 1 − sup {� x , u � : x ∈ K λ } = 1 − sup { r > 0 : ru ∈ � x ∈ K λ B ( x / 2 , � x � / 2) }

default Scaling limits B d − → 0 2 1 2 1 d +1 r λ (exp d − 1 ( λ − d +1 r λ (exp d − 1 ( λ − d +1 v )), ˆ d +1 v )) r λ ( v ) := λ ˆ s λ ( v ) := λ ( v ∈ R d − 1 ) exp d − 1 : R d − 1 ≃ T u 0 S d − 1 → S d − 1 exponential map at some fixed u 0 ∈ S d − 1 ˆ r λ and ˆ s λ converge in distribution in ( C ( B d − 1 (0 , R )) , �·� ∞ ), R > 0.

default Paraboloid growth process P := homogeneous Poisson point process on R d − 1 × R + Π ↑ := { ( v , h ) ∈ R d − 1 × R + : h ≥ � v � 2 / 2 } x ∈P ( x + Π ↑ ) Ψ := � The boundary ∂ Ψ is the limit of ˆ s λ .

default Paraboloid hull process P := homogeneous Poisson point process on R × R + Π ↓ := { ( v , h ) ∈ R d − 1 × R + : h ≤ −� v � 2 / 2 } x ∈ R d − 1 × R + : x +Π ↓ ∩P = ∅ ( x + Π ↓ ) Φ := � The boundary ∂ Φ is the limit of ˆ r λ .

default Central limit theorems and variance asymptotics f k ( K λ ) := number of k -faces of K λ V k ( K λ ):= k -th intrinsic volume of K λ (0 ≤ k ≤ d ) λ − d − 1 d +3 d +1 Var [ f k ( K λ )] → σ 2 d +1 Var [ V k ( K λ )] → σ 2 f k and λ V k . ( σ 2 f k , σ 2 V k explicit in terms of Φ and Ψ) f k ( K λ ) and V k ( K λ ) satisfy CLTs with convergence rates. Idea : each quantity can be written � x ∈P λ ξ ( x , P λ ) where ξ ( x , P λ ) = 0 if x not extreme and where ξ ( x , P λ ) depends on P λ only in a vicinity of x .

default Brownian limits B d − → 0