Convex hull of a random point set

Convex hull of a random point set Pierre Calka Journées nationales 26 GdR Informatique Mathématique Villetaneuse, 20 January 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 Joint works with Joseph Yukich (Lehigh University, USA) & Tomasz Schreiber (Toruń University, Poland)

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

Binomial point process 11 00 00 11 00 11 K convex body µ probability measure on K (X i,i 1) independent µ-distributed variables B4 0 0 1 1 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)

Poisson point process 11 00 B4 0 0 1 1 00 11 B1 B2 00 11 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

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

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

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

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

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

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

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)

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

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

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

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)

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

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)

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)

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

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

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

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

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

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

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

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.

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

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

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 )]).

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)

Dual characterization of extreme points

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.

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

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