Convex hull of a random point set Pierre Calka Journ ees - PowerPoint PPT Presentation

Convex hull of a random point set Pierre Calka Journ´ ees nationales 2016 GdR Informatique Math´ ematique Villetaneuse , 20 January 2016

default Outline Random polytopes: an overview Main results: variance asymptotics Case of the ball: sketch of proof and scaling limit Case of a simple polytope: sketch of proof and scaling limit Joint works with Joseph Yukich (Lehigh University, USA) & Tomasz Schreiber (Toru´ n University, Poland)

default Outline Random polytopes: an overview Poisson point process Uniform case Gaussian case Expectation asymptotics Main results: variance asymptotics Case of the ball: sketch of proof and scaling limit Case of a simple polytope: sketch of proof and scaling limit

default Binomial point process ◮ K convex body �� �� �� �� � � µ probability measure on K � �� �� � �� �� �� �� ( X i , i ≥ 1) independent µ -distributed variables � � � � �� �� B 4 B 1 � � � E n = { X 1 , · · · , X n } ( n ≥ 1) � � � �� �� �� �� �� �� ◮ Number of points in B 1 � B 2 � � B 3 � � #( E n ∩ B 1 ) binomial variable � � � � � � � �� �� � � n � µ ( B 1 ) k (1 − µ ( B 1 )) n − k , P (#( E n ∩ B 1 ) = k ) = k 0 ≤ k ≤ n ◮ #( E n ∩ B 1 ) , · · · , #( E n ∩ B ℓ ) not independent ( B 1 , · · · , B ℓ ∈ B ( R 2 ), B i ∩ B j = ∅ , i � = j )

default Poisson point process Poisson point process with intensity measure µ : �� �� locally finite subset P of R d such that �� �� � � � �� �� � �� �� �� �� ◮ #( P ∩ B 1 ) Poisson r.v. of mean µ ( B 1 ) � � � � �� �� B 4 B 1 P (#( P ∩ B 1 ) = k ) = e − µ ( B 1 ) µ ( B 1 ) k k ! , k ∈ N � � � � � � �� �� �� �� �� �� ◮ #( P ∩ B 1 ) , · · · , #( P ∩ B ℓ ) independent � � B 2 ( B 1 , · · · , B ℓ ∈ B ( R d ), B i ∩ B j = ∅ , i � = j ) � � B 3 � � � � � �� �� � If µ = λ d x , P said homogeneous of intensity λ

default Uniform case Binomial model K := convex body of R d ( X k , k ∈ N ∗ ):= independent and uniformly distributed in K K n := Conv( X 1 , · · · , X n ), n ≥ 1 K 50 , K ball K 50 , K square

default Uniform case Binomial model K := convex body of R d ( X k , k ∈ N ∗ ):= independent and uniformly distributed in K K n := Conv( X 1 , · · · , X n ), n ≥ 1 K 100 , K ball K 100 , K square

default Uniform case Binomial model K := convex body of R d ( X k , k ∈ N ∗ ):= independent and uniformly distributed in K K n := Conv( X 1 , · · · , X n ), n ≥ 1 K 500 , K ball K 500 , K square

default Uniform case Poisson model K := convex body of R d P λ , λ > 0 := Poisson point process of intensity measure λ d x K λ := Conv( P λ ∩ K ) K 500 , K ball K 500 , K square

default Gaussian case Poisson model (2 π ) d / 2 e −� x � 2 / 2 , x ∈ R d , d ≥ 2 1 ϕ d ( x ) := P λ , λ > 0 := Poisson point process of intensity measure λϕ d ( x ) d x K λ := Conv( P λ ) K 100 K 500

default Considered functionals ◮ f k ( · ): number of k -dimensional faces, 1 ≤ k ≤ d ◮ Vol ( · ): volume, V d − 1 ( · ): half-area of the boundary ◮ V k ( · ): k -th intrinsic volume, 1 ≤ k ≤ d The functionals V k are defined through Steiner formula: d � r d − k κ d − k V k ( K ) , where κ d := Vol ( B d ) Vol ( K + B (0 , r )) = k =0 d = 2: A ( K + B (0 , r )) = A ( K ) + P ( K ) r + π r 2

default Expectation asymptotics � � 1 − E Vol ( K n − 1 ) B. Efron’s relation (1965): E f 0 ( K n ) = n Vol ( K ) d − 1 1 Uniform case, K smooth E [ f k ( K λ )] ∼ d s λ � d +1 d +1 c d , k ∂ K κ s λ →∞ κ s := Gaussian curvature of ∂ K d , k F ( K ) log d − 1 ( λ ) Uniform case, K polytope E [ f k ( K λ )] ∼ c ′ λ →∞ F ( K ) := number of flags of K d − 1 2 ( λ ) Gaussian polytope E [ f k ( K λ )] ∼ d , k log c ′′ λ →∞ A. R´ enyi & R. Sulanke (1963), H. Raynaud (1970), R. Schneider & J. Wieacker (1978), F. Affentranger & R. Schneider (1992)

default Outline Random polytopes: an overview Main results: variance asymptotics Uniform case, K smooth Gaussian polytopes Uniform case, K simple polytope Case of the ball: sketch of proof and scaling limit Case of a simple polytope: sketch of proof and scaling limit

default Uniform case, K smooth: state of the art ◮ Identities relating higher moments C. Buchta (2005): d − 1 d +1 ≤ Var [ f 0 ( K λ )] c λ ◮ Number of faces and volume M. Reitzner (2005): d − 1 d − 1 d +1 ≤ Var [ f k ( K λ )] ≤ C λ c λ d +1 ◮ Intrinsic volumes I. B´ ar´ any, F. Fodor & V. Vigh (2009): c λ − d +3 d +1 ≤ Var [ V k ( K λ )] ≤ C λ − d +3 d +1 ◮ Central limit theorems M. Reitzner (2005): � t � � f k ( K λ ) − E [ f k ( K λ )] e − x 2 / 2 d x ≤ t → √ . P � Var [ f k ( K λ )] 2 π λ →∞ −∞

default Uniform case, K smooth: limiting variances K := convex body of R d with volume 1 and with a C 3 boundary κ := Gaussian curvature of ∂ K � λ →∞ λ − ( d − 1) / ( d +1) Var [ f k ( K λ )] = c k , d κ ( z ) 1 / ( d +1) dz lim ∂ K � λ →∞ λ ( d +3) / ( d +1) Var [ Vol ( K λ )] = c ′ κ ( z ) 1 / ( d +1) dz lim d ∂ K ( c k , d , c ′ d explicit positive constants) Remarks. ◮ Similar results for the binomial model ◮ Case of the ball: similar results for V k ( K λ ), functional central limit theorem for the defect volume

default Gaussian polytopes: state of the art ◮ Number of faces D. Hug & M. Reitzner (2005) , I. B´ ar´ any & V. Vu (2007): d − 1 d − 1 2 ( n ) ≤ Var [ f k ( K n )] ≤ C log 2 ( n ) c log ◮ Volume I. B´ ar´ any & V. Vu (2007): d − 3 d − 3 2 ( n ) ≤ Var [ Vol ( K n )] ≤ C log 2 ( n ) c log ◮ Central limit theorems I. B´ ar´ any & V. Vu (2007) ◮ Intrinsic volumes D. Hug & M. Reitzner (2005): k − 3 2 ( n ) Var [ V k ( K n )] ≤ C log

default Gaussian polytopes: limiting variances n →∞ log − d − 1 2 ( λ ) Var [ f k ( K λ )] = c d , k ∈ (0 , ∞ ) lim n →∞ log − k + d +3 2 ( λ ) Var [ V k ( K λ )] = c ′ lim d , k ∈ [0 , ∞ ) λ →∞ 1 − k log(log λ ) − 1 log − k / 2 ( λ ) E [ V k ( K λ )] c ′′ + O ((log − 1 ( λ )) = d , k 4 log λ Remarks. ◮ Similar results for the binomial model ◮ Intrinsic volumes: for 1 ≤ k ≤ ( d − 1), c ′ d , k ∈ [0 , ∞ ) ◮ Functional CLT for the defect volume

default Uniform case, K polytope: state of the art ◮ Number of faces and volume I. B´ ar´ any & M. Reitzner (2010) d , k F ( K ) 3 log d − 1 ( λ ) c d , k F ( K ) log d − 1 ( λ ) ≤ Var [ f k ( K λ )] ≤ c ′ c d , k F ( K )log d − 1 ( λ ) d , k F ( K ) 3 log d − 1 ( λ ) ≤ Var [ Vol ( K λ )] ≤ c ′ λ 2 λ 2 ◮ Central limit theorems I. B´ ar´ any & M. Reitzner (2010b)

default Uniform case, K simple polytope: limiting variances K := simple polytope of R d with volume 1 λ →∞ log − ( d − 1) ( λ ) Var [ f k ( K λ )] = c d , k f 0 ( K ) lim λ →∞ λ 2 log − ( d − 1) ( λ ) Var [ Vol ( K λ )] = c ′ lim d , k f 0 ( K )

default Outline Random polytopes: an overview Main results: variance asymptotics Case of the ball: sketch of proof and scaling limit Calculation of the variance of f k ( K λ ) Scaling transform Dual characterization of extreme points Action of the scaling transform Case of a simple polytope: sketch of proof and scaling limit

default Calculation of the expectation of f k ( K λ ) ◮ Decomposition:    � E [ f k ( K λ )] = E ξ ( x , P λ )  x ∈P λ 1 � k +1 # k -face containing x if x extreme ξ ( x , P λ ) := 0 if not ◮ Mecke-Slivnyak formula � E [ f k ( K λ )] = λ B d E [ ξ ( x , P λ ∪ { x } )] d x

default Calculation of the variance of f k ( K λ ) Var [ f k ( K λ )]    − ( E [ f k ( K λ )]) 2  � ξ 2 ( x , P λ ) + � = E ξ ( x , P λ ) ξ ( y , P λ ) x ∈P λ x � = y ∈P λ � B d E [ ξ 2 ( x , P λ ∪ { x } )] d x = λ �� + λ 2 ( B d ) 2 E [ ξ ( x , P λ ∪ { x , y } ) ξ ( y , P λ ∪ { x , y } )] d x d y �� − λ 2 ( B d ) 2 E [ ξ ( x , P λ ∪ { x } )] E [ ξ ( y , P λ ∪ { y } )] d x d y � B d E [ ξ 2 ( x , P λ ∪ { x } )] d x = λ �� + λ 2 ( B d ) 2 ” Cov” ( ξ ( x , P λ ∪ { x } ) , ξ ( y , P λ ∪ { y } )) d x d y

default Scaling transform Question : Limits of E [ ξ ( x , P λ )] and ” Cov” ( ξ ( x , P λ ) , ξ ( y , P λ ))? Answer : definition of limit scores in a new space ◮ Scaling transform : B d − 1 \ { 0 } R d − 1 × R + � − → T λ : 1 2 d +1 exp − 1 d +1 (1 − � x � )) x �− → ( λ d − 1 ( x / � x � ) , λ exp d − 1 : R d − 1 ≃ T u 0 S d − 1 → S d − 1 exponential map at u 0 ∈ S d − 1 ◮ Image of a score : ξ ( λ ) ( T λ ( x ) , T λ ( P λ )) := ξ ( x , P λ ) ◮ Convergence of P λ : T λ ( P λ ) D → P where P := homogeneous Poisson point process in R d − 1 × R of intensity 1