Convex hull of a random point set

Transcription

1 Convex hull of a random point set Pierre Calka Journées nationales 26 GdR Informatique Mathématique Villetaneuse, 20 January 26

2 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 (Toruń University, Poland)

3 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

4 Binomial point process K convex body µ probability measure on K (X i,i 1) independent µ-distributed variables B B1 E n = {X 1,,X n } (n 1) B2 B3 Number of points in B 1 #(E n B 1 ) binomial variable P(#(E n B 1 ) = k) = ( n k) µ(b1 ) k (1 µ(b 1 )) n 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)

5 Poisson point process B B1 B 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 ) P(#(P B 1 ) = k) = e µ(b1) µ(b 1) k k!, k N #(P B 1 ),,#(P B l ) independent B3 (B 1,,B l B(R d ), B i B j =, i j) If µ = λdx, P said homogeneous of intensity λ

6 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

7 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

8 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

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

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

11 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 Vol(K+B(0,r)) = r d k κ d k V k (K), where κ d := Vol(B d ) k=0 d = 2: A(K +B(0,r)) = A(K)+P(K)r +πr 2

12 Expectation asymptotics ( B. Efron s relation (1965): Ef 0 (K n ) = n 1 EVol(K ) n 1) Vol(K) Uniform case, K smooth κ s := Gaussian curvature of K E[f k (K λ )] λ c d,k s ds λ d 1 d+1 K κ 1 d+1 Uniform case, K polytope E[f k (K λ )] λ c d,k F(K) logd 1 (λ) F(K) := number of flags of K Gaussian polytope E[f k (K λ )] c d,k log d 1 2 (λ) λ A. Rényi & R. Sulanke (1963), H. Raynaud (1970), R. Schneider & J. Wieacker (1978), F. Affentranger & R. Schneider (1992)

13 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

14 Uniform case, K smooth: state of the art Identities relating higher moments C. Buchta (2005): Number of faces and volume M. Reitzner (2005): Intrinsic volumes I. Bárány, F. Fodor & V. Vigh (2009): Central limit theorems M. Reitzner (2005): P cλ d 1 d+1 Var[f0 (K λ )] cλ d 1 d+1 Var[fk (K λ )] Cλ d 1 d+1 cλ d+3 d+1 Var[Vk (K λ )] Cλ d+3 d+1 [ f k (K λ ) E[f k (K λ )] t Var[fk (K λ )] ] t e x2 /2 dx. λ 2π

15 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 lim λ λ (d 1)/(d+1) Var[f k (K λ )] = c k,d κ(z) 1/(d+1) dz K lim λ λ(d+3)/(d+1) Var [Vol(K λ )] = c d κ(z) 1/(d+1) dz K (ck,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

16 Gaussian polytopes: state of the art Number of faces D. Hug & M. Reitzner (2005), I. Bárány & V. Vu (2007): Volume I. Bárány & V. Vu (2007): Central limit theorems I. Bárány & V. Vu (2007) Intrinsic volumes D. Hug & M. Reitzner (2005): clog d 1 2 (n) Var[f k (K n )] C log d 1 2 (n) clog d 3 2 (n) Var[Vol(K n )] C log d 3 2 (n) Var[V k (K n )] C log k 3 2 (n)

17 Gaussian polytopes: limiting variances lim log d 1 2 (λ)var[f k (K λ )] = c d,k (0, ) n lim log k+d+3 2 (λ)var[vk (K λ )] = c n d,k [0, ) c d,k 1 log k/2 (λ)e[v k (K λ )] = 1 klog(logλ) +O((log 1 (λ)) λ 4logλ Remarks. Similar results for the binomial model Intrinsic volumes: for 1 k (d 1), c d,k [0, ) Functional CLT for the defect volume

18 Uniform case, K polytope: state of the art Number of faces and volume I. Bárány & M. Reitzner (20) c d,k F(K)log d 1 (λ) Var[f k (K λ )] c d,kf(k) 3 log d 1 (λ) c d,k F(K) logd 1 (λ) λ 2 Var[Vol(K λ )] c d,k F(K)3logd 1 (λ) λ 2 Central limit theorems I. Bárány & M. Reitzner (20b)

19 Uniform case, K simple polytope: limiting variances K := simple polytope of R d with volume 1 lim λ log (d 1) (λ)var[f k (K λ )] = c d,k f 0 (K) lim λ λ2 log (d 1) (λ)var[vol(k λ )] = c d,k f 0(K)

20 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

21 Calculation of the expectation of f k (K λ ) Decomposition: E[f k (K λ )] = E ξ(x,p λ ) x P λ ξ(x,p λ ) := { 1 k+1 #k-face containing x if x extreme 0 if not Mecke-Slivnyak formula E[f k (K λ )] = λ E[ξ(x,P λ {x})]dx B d

22 Calculation of the variance of f k (K λ ) Var[f k (K λ )] = E ξ 2 (x,p λ )+ ξ(x,p λ )ξ(y,p λ ) (E[f k (K λ )]) 2 x P λ x y P λ = λ E[ξ 2 (x,p λ {x})]dx B d +λ (B 2 E[ξ(x,P λ {x,y})ξ(y,p λ {x,y})]dxdy d ) 2 λ 2 (B d ) 2 E[ξ(x,P λ {x})]e[ξ(y,p λ {y})]dxdy = λ E[ξ 2 (x,p λ {x})]dx B d +λ 2 (B d ) 2 Cov (ξ(x,p λ {x}),ξ(y,p λ {y}))dxdy

23 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 T λ d 1 \{0} R d 1 R + : x (λ 1 d+1 exp 1 d 1 (x/ x ),λ 2 d+1 (1 x )) exp d 1 : R d 1 T u0 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

24 Dual characterization of the extreme points Given K λ contains the origin, x P λ extreme H support hyperplane of K λ, x H ( x y B 2, x ) s. t. 0 and P λ \{x} on the same side of (x +y ) 2 ( x the petal of x, B 2, x ) ( ) x B 2 2, x 2 x P λ \{x} B d 0

25 Action of the scaling transform Π := {(v,h) R d 1 R : h v 2 2 }, Π := {(v,h) R d 1 R : h v 2 2 } Half-space Translate of Π Boundary of the convex hull Union of portions of down paraboloids Petal Translate of Π Extreme point (x +Π ) not fully covered B d 0

26 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 Floating body Additivity of the variance over the vertices Scaling transform in the vicinity of a vertex Dual characterization of extreme points Action of the scaling transform

27 Floating body v(x) := inf{vol(k H + ) : H + half-space containing x}, x K Floating body : K(v t) := {x K : v(x) t} K(v t) is a convex body and K(v 1/λ) is close to K λ.

28 Floating body v(x) := inf{vol(k H + ) : H + half-space containing x}, x K Floating body : K(v t) := {x K : v(x) t} K(v t) is a convex body and K(v 1/λ) is close to K λ. B d (v 1/λ) = (1 f(λ))b d f(λ) cλ 2 d+1

29 Comparison between K λ and the floating body Expectation Bárány & Larman (1988): cvol(k(v 1/λ)) Vol(K) E[Vol(K λ )] CVol(K(v 1/λ)) Variance Bárány & Reitzner (20): Sandwiching (polytope case) Bárány & Reitzner (20b): s := cλ 1 Vol(K(v 1/λ)) Var[Vol(K λ )] P[ K λ [K(v s)\k(v T)]] = O ((log(λ)) 4d2) c, T := λ(log(λ)) 4d2 +d 1 c log(log(λ)) λ

30 Additivity of the variance over the vertices V(K) := set of vertices of K p δ (v) := parallelepiped with volume δ d at v where δ = exp( (log 1 d(λ))) Z v := (k +1) 1 x P λ p δ (v)#{k-faces containing x} Var[f k (K λ )] = v V(K) Var[Z v ]+o(var[f k (K λ )]).

31 Scaling transform in the vicinity of a vertex K identified with (0, ) d after scaling transformation Floating body K(v = t λ ) = {(z 1,,z d ) (0, ) d : d i=1 z i = c t λ } V := {(y 1,,y d ) R d : d i=1 y i = 0} = R d 1 Scaling transform: { (0, ) d V R T (λ) : ( ) (z 1,,z d ) proj V (log(z)), 1 d log(λ d i=1 z i)

32 Scaling transform in the vicinity of a vertex K identified with (0, ) d after scaling transformation Floating body K(v = t λ ) = {(z 1,,z d ) (0, ) d : d i=1 z i = c t λ } V := {(y 1,,y d ) R d : d i=1 y i = 0} = R d 1 Scaling transform: { (0, ) d V R T (λ) : ( ) (z 1,,z d ) proj V (log(z)), 1 d log(λ d i=1 z i)

33 Scaling transform in the vicinity of a vertex K identified with (0, ) d after scaling transformation Floating body K(v = t λ ) = {(z 1,,z d ) (0, ) d : d i=1 z i = c t λ } V := {(y 1,,y d ) R d : d i=1 y i = 0} = R d 1 Scaling transform: { (0, ) d V R T (λ) : ( ) (z 1,,z d ) proj V (log(z)), 1 d log(λ d i=1 z i)

34 Scaling transform in the vicinity of a vertex K identified with (0, ) d after scaling transformation Floating body K(v = t λ ) = {(z 1,,z d ) (0, ) d : d i=1 z i = c t λ } V := {(y 1,,y d ) R d : d i=1 y i = 0} = R d 1 Scaling transform: { (0, ) d V R T (λ) : ( ) (z 1,,z d ) proj V (log(z)), 1 d log(λ d i=1 z i) Convergence of P λ : T λ (P λ ) D P where P := Poisson point process in R d 1 R of intensity measure de dh dvdh

35 Dual characterization of extreme points

36 Dual characterization of extreme points

37 Dual characterization of extreme points

38 Dual characterization of extreme points

39 Dual characterization of extreme points Each point z (0, ) d generates a petal S(z), i.e. the set of all tangency points of surfaces K(v = t λ ), t > 0, with the hyperplanes containing z. z is cone-extreme iff S(z) is not fully covered by the other petals.

40 Action of the scaling transform G(v) := log( 1 ) d d k=1 el k(v), v = (l 1 (v),,l d (v)) V Π := {(v,h) R d 1 R : h G( v)}, Π := {(v,h) R d 1 R : h G(v)} Floating bodies horizontal half-spaces Boundary of the convex hull Union of portions down cone-like grains Petal Translate of Π Extreme point (x +Π ) not fully covered

41 Thank you for your attention!