Convex discretization of functionals involving the Monge-Amp`ere operator

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Convex discretization of functionals involving the Monge-Amp`ere operator

Quentin M´erigot

CNRS / Universit´e Paris-Dauphine

Joint work with J.D. Benamou, G. Carlier and ´E. Oudet

Workshop on Optimal Transport in the Applied Sciences

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1. Motivation: Gradient flows in Wasserstein space

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Background: Optimal transport

I Wasserstein distance between µ, ν ∈ P₂(R^d), P₂(R^d) = {µ ∈ P(R^d); R

kxk² d µ(x) < +∞}

Γ(µ, ν) := {π ∈ P(R^d × R^d); p_1#π = µ, p_2#π = ν} P₂^ac(R^d) = P₂(R^d) ∩ L¹(R^d)

R^d

R^d π

µ ν

p₁ p₂

Definition: W²₂(µ, ν) := min_{π∈Γ(µ,ν}₎ R

kx − yk² d π(x, y).

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Background: Optimal transport

kxk² d µ(x) < +∞}

R^d

R^d π

µ ν

p₁ p₂

I Relation to convex functions: Def: K := finite convex functions on R^d Definition: W²₂(µ, ν) := min_{π∈Γ(µ,ν}₎ R

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Background: Optimal transport

Theorem (Brenier): Given µ ∈ P₂^ac(R^d) the map φ ∈ K 7→ ∇φ_#µ ∈ P₂(R^d) is surjective and moreover,

kxk² d µ(x) < +∞}

R^d

R^d π

µ ν

p₁ p₂

[Brenier ’91]

W²₂(µ, ∇φ_#µ) = R

R^d kx − ∇φ(x)k² d µ(x)

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Background: Optimal transport

Theorem (Brenier): Given µ ∈ P₂^ac(R^d) the map φ ∈ K 7→ ∇φ_#µ ∈ P₂(R^d) is surjective and moreover,

kxk² d µ(x) < +∞}

R^d

R^d π

µ ν

p₁ p₂

[Brenier ’91]

Given any µ ∈ P₂^ac(R^d), we get a ”parameterization” of P₂(R^d), as ”seen” from µ.

W²₂(µ, ∇φ_#µ) = R

R^d kx − ∇φ(x)k² d µ(x)

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Motivation 1: Crowd Motion Under Congestion

[Maury-Roudneff-Chupin-Santambrogio 10]

ρ^τ_k+1 = min_σ∈P₂_(X₎ _2τ¹ W²₂(ρ^τ_k, σ) + E(σ) + U(σ) I JKO scheme for crowd motion with hard congestion:

U(ν) :=

( 0 if d ν = f d H^d, f ≤ 1

+ ∞ if not congestion potential energy

E(ν) := R

R^d V (x) d ν(x) (∗)

where X ⊆ R^d is convex and bounded, and

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Motivation 1: Crowd Motion Under Congestion

[Maury-Roudneff-Chupin-Santambrogio 10]

min_φ _2τ¹ R

R^d kx − ∇φ(x)k²ρ^τ_k(x) d x + E(∇φ_#ρ^τ_k) + U(∇φ_#ρ^τ_k) ρ^τ_k+1 = min_σ∈P₂_(X₎ _2τ¹ W²₂(ρ^τ_k, σ) + E(σ) + U(σ)

I JKO scheme for crowd motion with hard congestion:

U(ν) :=

( 0 if d ν = f d H^d, f ≤ 1

+ ∞ if not congestion potential energy

E(ν) := R

R^d V (x) d ν(x)

I Assuming σ = ∇φ_#ρ^τ_k with φ convex, the Wasserstein term becomes explicit:

On the other hand, the constraint becomes strongly nonlinear:

(∗)

(∗) ⇐⇒

U(∇φ ρ^τ) < +∞ ⇐⇒ det D²φ(x) ≥ ρ (x) where X ⊆ R^d is convex and bounded, and

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Motivation 2: Nonlinear Diffusion

∂ρ

∂t = div [ρ∇(U⁰(ρ) + V + W ∗ ρ)] ρ(0, .) = ρ₀

(∗) ρ(t, .) ∈ P^ac(R^d)

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Motivation 2: Nonlinear Diffusion

∂ρ

∂t = div [ρ∇(U⁰(ρ) + V + W ∗ ρ)] ρ(0, .) = ρ₀ (∗)

I Formally, (∗) can be seen as the W₂-gradient flow of U + E, with

ρ(t, .) ∈ P^ac(R^d)

U(ν) :=





 Z

R^d

U(f(x)) d x if d ν = f d H^d + ∞ if not

E(ν) := R

R^d V (x) d ν(x) + R

R^d W (x − y) d[ν ⊗ ν](x, y)

internal energy, ex: U(r) = r log r =⇒ entropy

potential energy interaction energy

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Motivation 2: Nonlinear Diffusion

∂ρ

∂t = div [ρ∇(U⁰(ρ) + V + W ∗ ρ)]

I JKO time discrete scheme: for τ > 0,

ρ(0, .) = ρ₀ (∗)

ρ^τ_k+1 = arg min_σ∈P(_R^d₎ _2τ¹ W₂(ρ^τ_k, σ)² + U(σ) + E(σ) I Formally, (∗) can be seen as the W₂-gradient flow of U + E, with

ρ(t, .) ∈ P^ac(R^d)

[Jordan-Kinderlehrer-Otto ’98]

U(ν) :=





 Z

R^d

E(ν) := R

R^d W (x − y) d[ν ⊗ ν](x, y)

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Motivation 2: Nonlinear Diffusion

∂ρ

∂t = div [ρ∇(U⁰(ρ) + V + W ∗ ρ)]

I JKO time discrete scheme: for τ > 0,

ρ(0, .) = ρ₀ (∗)

ρ^τ_k+1 = arg min_σ∈P(_R^d₎ _2τ¹ W₂(ρ^τ_k, σ)² + U(σ) + E(σ)

−→ Many applications: porous medium equation, cell movement via chemotaxis, I Formally, (∗) can be seen as the W₂-gradient flow of U + E, with

ρ(t, .) ∈ P^ac(R^d)

[Jordan-Kinderlehrer-Otto ’98]

U(ν) :=





 Z

R^d

E(ν) := R

R^d W (x − y) d[ν ⊗ ν](x, y)

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Displacement Convex Setting

min_ν_∈P_(X₎ _2τ¹ W²₂(µ, ν) + E(ν) + U(ν) For X convex bounded and µ ∈ P^ac(X),

spt(ν) ⊆ X

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Displacement Convex Setting

min_ν_∈P_(X₎ _2τ¹ W²₂(µ, ν) + E(ν) + U(ν) For X convex bounded and µ ∈ P^ac(X),

(∗X)

min_φ∈K_X _2τ¹ W²₂(µ, ∇φ_#µ) + U(∇φ_#µ) + E(∇φ_#µ)

⇐⇒

K_X := {φ convex ; ∇φ ∈ X}

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Displacement Convex Setting

When is the minimization problem (∗_X) convex ?

E(ν) := R

R^d(V + W ∗ ν) d ν U(ν) := R

R^d U _d^d_H^ν_d

d x min_ν_∈P_(X₎ _2τ¹ W²₂(µ, ν) + E(ν) + U(ν)

For X convex bounded and µ ∈ P^ac(X),

(∗X)

⇐⇒

K_X := {φ convex ; ∇φ ∈ X}

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Displacement Convex Setting

E(ν) := R

R^d(V + W ∗ ν) d ν U(ν) := R

R^d U _d^d_H^ν_d

(∗X)

⇐⇒

NB: U(∇φ_#ρ) = R

U

ρ(x) MA[φ](x)

MA[φ](x) d x MA[φ](x) := det(D²φ(x)) K_X := {φ convex ; ∇φ ∈ X}

(H2) r^dU(r^−d) is convex non-increasing, U(0) = 0.

Theorem: (∗_X) is convex if (H1) V, W : R^d → R are convex functions,

[McCann ’94]

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Displacement Convex Setting

E(ν) := R

R^d(V + W ∗ ν) d ν U(ν) := R

R^d U _d^d_H^ν_d

(∗X)

⇐⇒

NB: U(∇φ_#ρ) = R

U

ρ(x) MA[φ](x)

MA[φ](x) d x MA[φ](x) := det(D²φ(x)) K_X := {φ convex ; ∇φ ∈ X}

(H2) r^dU(r^−d) is convex non-increasing, U(0) = 0.

Theorem: (∗_X) is convex if (H1) V, W : R^d → R are convex functions,

[McCann ’94]

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Numerical applications of the JKO scheme

I Numerical applications of the (variational) JKO scheme are still limited:

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Numerical applications of the JKO scheme

1D: monotone rearrangement e.g. [Kinderleherer-Walkington 99]

[Blanchet-Calvez-Carrillo 08] [Agueh-Bowles 09]

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Numerical applications of the JKO scheme

2D: optimal transport plans → diffeomorphisms [Carrillo-Moll 09]

U = hard congestion term [Maury-Roudneff-Chupin-Santambrogio 10]

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Numerical applications of the JKO scheme

I Our goal is to approximate a JKO step numerically, in dimension ≥ 2:

min_ν_∈P_(X₎ _2τ¹ W²₂(µ, ν) + E(ν) + U(ν)

(∗_X)

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Numerical applications of the JKO scheme

I Our goal is to approximate a JKO step numerically, in dimension ≥ 2:

Part 2: Convex discretization of the problem (∗_X) under McCann’s hypotheses.

Part 3: Γ-convergence results from the discrete problem to the continuous one.

Part 4: Numerical simulations: non-linear diffusion, crowd motion.

min_ν_∈P_(X₎ _2τ¹ W²₂(µ, ν) + E(ν) + U(ν)

(∗_X)

I Plan of the talk:

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2. Convex discretization of a JKO step

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Prior work: Discretizing the Space of Convex Functions

min_φ∈K R

X F (φ(x), ∇φ(x)) d µ(x)

I Finite elements: piecewise-linear convex functions K := {φ convex}

−→ number of constraints is ∼ N := card(P )

P ⊆ X

over a fixed mesh

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Prior work: Discretizing the Space of Convex Functions

min_φ∈K R

X F (φ(x), ∇φ(x)) d µ(x)

[Chon´e-Le Meur ’99]

I Finite elements: piecewise-linear convex functions K := {φ convex}

−→ non-convergence result

P ⊆ X

over a fixed mesh

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Prior work: Discretizing the Space of Convex Functions

−→ convergent, but number of constraints is ' N²

[Ekeland—Moreno-Bromberg ’10]

min_φ∈K R

X F (φ(x), ∇φ(x)) d µ(x)

I Finite elements: piecewise-linear convex functions

I Finite differences, using convex interpolates

K := {φ convex}

[Carlier-Lachand-Robert-Maury ’01]

P ⊆ X

over a fixed mesh

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Prior work: Discretizing the Space of Convex Functions

[Mirebeau ’14]

min_φ∈K R

X F (φ(x), ∇φ(x)) d µ(x)

[Oudet-M. ’14]

[Oberman ’14]

−→ exterior parameterization

K := {φ convex}

−→ adaptive method

P ⊆ X

over a fixed mesh

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Prior work: Discretizing the Space of Convex Functions

[Mirebeau ’14]

min_φ∈K R

X F (φ(x), ∇φ(x)) d µ(x)

[Oudet-M. ’14]

[Oberman ’14]

−→ exterior parameterization

K := {φ convex}

−→ adaptive method

P ⊆ X

over a fixed mesh

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Discretizing the Space of Convex Functions

min_φ∈K_X _2τ¹ W²₂(µ, ∇φ_#µ) + E(∇φ_#µ) + U(∇φ_#µ) K_X := {φ cvx ; ∇φ ∈ X}

Definition: Given P ⊆ X finite, K_X(P ) = {ψ|_P ; ψ ∈ K_X}

= φ : P → R

P ⊆ X

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Discretizing the Space of Convex Functions

−→ finite-dimensional convex set

= φ : P → R

P ⊆ X

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Discretizing the Space of Convex Functions

−→ extension of φ ∈ K_X(P ): ⁼ ^φ ^: ^P ^→ _R

φˆ

P ⊆ X

φˆ := max{ψ; ψ ∈ K_X and ψ|_P ≤ φ} ∈ K_X

φ : P → R φˆ : R^d → R ∈ KX

X = [−1, 1]

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Discretizing the Space of Convex Functions

φˆ

P ⊆ X

Discrete push-forward of µ_P = P

p∈P µ_pδ_p ∈ P(P)

φ : P → R φˆ : R^d → R ∈ KX

X = [−1, 1]

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Discretizing the Space of Convex Functions

φˆ

P ⊆ X

p

p∈P µ_pδ_p ∈ P(P) Definition: For φ ∈ K_X(P ), let V_p := ∂φ(p)ˆ and

φ : P → R φˆ : R^d → R ∈ KX

X = [−1, 1]

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Discretizing the Space of Convex Functions

φˆ

P ⊆ X

X Gφ_#µ_P := P

p∈P

µ_p

H^d(V_p) H^d

V_p ∈ P^ac(X) p

Vp

p∈P µ_pδ_p ∈ P(P) Definition: For φ ∈ K_X(P ), let V_p := ∂φ(p)ˆ and

φ : P → R φˆ : R^d → R ∈ KX

X = [−1, 1]

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Convex Space-Discretization of One JKO Step

Theorem: Under McCann’s hypothesis (H1) and (H2), the problem min_φ∈K^G

X(P) 1

2τ W²₂(µ_P , Hφ_#µ_P ) + E(Hφ_#µ_P ) + U(Gφ_#µ_P ) is convex, and the minimum is unique if r^dU(r^−d) is strictly convex.

[Benamou, Carlier, M., Oudet ’14]

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Convex Space-Discretization of One JKO Step

X(P) 1

I Unfortunately, two different definitions of push-forward seem necessary.

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Convex Space-Discretization of One JKO Step

X(P) 1

of the discrete Monge-Amp`ere operator:

I The convexity of φ ∈ K_X(P ) 7→ U(Gφ_#µ_P) follows from the log-concavity MA[φ](p) := H^d(∂φ(p)).ˆ

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Convex Space-Discretization of One JKO Step

X(P) 1

U(Gφ_#µ_P) < +∞ =⇒ ∀p ∈ P, MA[φ](p) = H^d(∂φ(p))ˆ > 0

I If lim_r→∞ U(r)/r = +∞, the internal energy is a barrier for convexity, i.e.

=⇒ φ is in the interior of K_X(P ).

2

of the discrete Monge-Amp`ere operator:

I The convexity of φ ∈ K_X(P ) 7→ U(Gφ_#µ_P) follows from the log-concavity MA[φ](p) := H^d(∂φ(p)).ˆ

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Convex Space-Discretization of the Internal Energy

Proposition: Under McCann’s assumption, φ ∈ K_X(P ) 7→ U(Gφ_#µ_P ) is convex.

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Convex Space-Discretization of the Internal Energy

Discrete Monge-Amp`ere operator: MA[φ](p) := H^d(∂φ(p)).ˆ

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Convex Space-Discretization of the Internal Energy

U(G^ac_φ#µ_P ) = P

p∈P U (µ_p/MA[φ](p)) MA[φ](p)

Proof: Recall Gφ_#µ_P = P

p∈P [µ_p/MA[φ](p)]1_∂_φ(p)_ˆ . With U(σ) = R

U(σ(x)) d x, Discrete Monge-Amp`ere operator: MA[φ](p) := H^d(∂φ(p)).ˆ

NB: similarity with U(∇φ_#ρ) = R

U

ρ(x) MA[φ](x)

MA[φ](x) d x

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Convex Space-Discretization of the Internal Energy

U(σ(x)) d x,

= − P

p∈P µ_p log(MA[φ](p)) + P

p∈P µ_p log(µ_p) case U(r) = r log r

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Convex Space-Discretization of the Internal Energy

U(σ(x)) d x,

= − P

p∈P µ_p log(µ_p)

Lemma: Given φ₀, φ₁ ∈ K_X(P ), φ_t := (1 − t)φ₀ + tφ₁ and

∂φˆ_t(p) ⊇ (1 − t)∂φˆ₀(p) ⊕ t∂φˆ₁(p).

case U(r) = r log r

⊕ = Minkowski sum Discrete Monge-Amp`ere operator: MA[φ](p) := H^d(∂φ(p)).ˆ

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Convex Space-Discretization of the Internal Energy

U(σ(x)) d x,

= − P

log H^d(∂φˆ_t) ≥ log H^d((1 − t)∂φˆ₀(p) ⊕ t∂φˆ₁(p)).

≥ (1 − t) log H^d(∂φˆ₀(p)) + t log H^d( ˆφ₁(p))).

Brunn-Minkowski

∂φˆ_t(p) ⊇ (1 − t)∂φˆ₀(p) ⊕ t∂φˆ₁(p).

case U(r) = r log r

⊕ = Minkowski sum

=⇒

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Convex Space-Discretization of the Internal Energy

U(σ(x)) d x,

= − P

log H^d(∂φˆ_t) ≥ log H^d((1 − t)∂φˆ₀(p) ⊕ t∂φˆ₁(p)).

≥ (1 − t) log H^d(∂φˆ₀(p)) + t log H^d( ˆφ₁(p))).

Brunn-Minkowski

∂φˆ_t(p) ⊇ (1 − t)∂φˆ₀(p) ⊕ t∂φˆ₁(p).

case U(r) = r log r

⊕ = Minkowski sum

=⇒

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3. A Γ-convergence result

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Convergence of the Space-Discretization

Setting: X bounded and convex, µ ∈ P^ac(X) with density c⁻¹ ≤ ρ ≤ c min_ν_∈P_(X₎ _2τ¹ W²₂(µ, ν) + E(ν) + U(ν) (∗)

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Convergence of the Space-Discretization

Setting: X bounded and convex, µ ∈ P^ac(X) with density c⁻¹ ≤ ρ ≤ c

(C1) E continuous, U l.s.c on (P(X), W₂)

min_ν_∈P_(X₎ _2τ¹ W²₂(µ, ν) + E(ν) + U(ν) (∗)

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Convergence of the Space-Discretization

min_ν_∈P_(X₎ _2τ¹ W²₂(µ, ν) + E(ν) + U(ν) (∗)

(C2) U(ρ) = R

U(ρ(x)) d x, where U ≥ M is convex.

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Convergence of the Space-Discretization

min_ν_∈P_(X₎ _2τ¹ W²₂(µ, ν) + E(ν) + U(ν) (∗)

(C2) U(ρ) = R

6= McCann’s condition for displacement-convexity

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Convergence of the Space-Discretization

min_ν_∈P_(X₎ _2τ¹ W²₂(µ, ν) + E(ν) + U(ν) (∗)

Theorem: Let P_n ⊆ X finite, µ_n ∈ P(P_n) with lim W₂(µ_n, µ) = 0, and:

min_φ∈K^G

X(P_n) 1

2τ W₂(µ_n, Hφ_#µ_n) + E(Hφ_#µ_n) + U(Gφ_#µ_n) (∗)_n If φ_n minimizes (∗)_n, then (Gφ_n_#µ_n) is a minimizing sequence for (∗).

(C2) U(ρ) = R

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Convergence of the Space-Discretization

min_ν_∈P_(X₎ _2τ¹ W²₂(µ, ν) + E(ν) + U(ν) (∗)

min_φ∈K^G

X(P_n) 1

(C2) U(ρ) = R

I Proof relies on Caffarelli’s regularity theorem.

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Convergence of the Space-Discretization

min_ν_∈P_(X₎ _2τ¹ W²₂(µ, ν) + E(ν) + U(ν) (∗)

min_φ∈K^G

X(P_n) 1

(C2) U(ρ) = R

I Proof relies on Caffarelli’s regularity theorem.

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Proof of the convergence theorem: Lower bound

Let m := min_σ∈P_(X₎ F(σ)

m_n := min_φ_n_∈K_X_(P_n₎ F_n(Gφ_n_#µ_n) I Step 1: lim inf m_n ≥ m

JKO step: X bounded and convex, µ ∈ P^ac(X) with density c⁻¹ ≤ ρ ≤ c

F(σ) := W²₂(µ, σ) + E(σ) + U(σ) F_n(σ) := W²₂(µ_n, σ) + E(σ) + U(σ)

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Proof of the convergence theorem: Lower bound

I Step 2: lim sup m_n ≤ m = min_σ∈P_(X₎ F(σ)

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Proof of the convergence theorem: Lower bound

I Step 2: lim sup m_n ≤ m = min_σ∈P_(X₎ F(σ)

s.t. lim_n→∞ kσ − σ_nk_L^∞_(X₎ = 0, where σ_n is the density of Gφ_n_#µ_n.

Using (C2) and a convolution argument

⇐= Given any probability density σ ∈ C⁰(X) with ε < σ < ε⁻¹, ∃φ_n ∈ K_X(P_n)

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Proof of the convergence theorem: Lower bound

X X

∇φ

µ σ = ∇φ_#µ

I Step 2’:

s.t. lim_n→∞ kσ − σ_nk_L^∞_(X₎ = 0, where σ_n := density of Gφ_n_#µ_n. Given any prob. density σ ∈ C⁰(X) s.t. ε < σ < ε⁻¹, ∃φ_n ∈ K_X(P_n)

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Proof of the convergence theorem: Lower bound

X X X

spt(µ ) = P ⊆ X

∇φ

µ σ = ∇φ_#µ

I Step 2’:

∀p ∈ P_n, σ(V_pⁿ) = µ_p, where V_pⁿ := ∂φˆ_n(p)

∂φˆn

(a) Given µ_n = P

p∈P_n µ_pδ_p, take φ_n ∈ K_X(P_n) O.T. potential between µ_n and σ:

p V_pⁿ

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Proof of the convergence theorem: Lower bound

X X X

∇φ

µ σ = ∇φ_#µ

I Step 2’:

∀p ∈ P_n, σ(V_pⁿ) = µ_p, where V_pⁿ := ∂φˆ_n(p) (b) σ_n := Gφ_n_#µ_n = P

p∈P_n

σ(V_pⁿ)

H^d(V_pⁿ)1_V ⁿ

p .

∂φˆn

(a) Given µ_n = P

p V_pⁿ

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Proof of the convergence theorem: Lower bound

X X X

spt(µ ) = P ⊆ X

∇φ

µ σ = ∇φ_#µ

I Step 2’:

p∈P_n

σ(V_pⁿ)

H^d(V_pⁿ)1_V ⁿ

p . (c) If max_p∈P_n diam(V_pⁿ) −−−→^n→∞ 0 , then

∂φˆn

(a) Given µ_n = P

p V_pⁿ

lim_n→∞ kσ − σ_nk_L^∞_(X₎ = 0.

(1)

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Proof of the convergence theorem: Lower bound

X X X

∇φ

µ σ = ∇φ_#µ

I Step 2’:

p∈P_n

σ(V_pⁿ)

H^d(V_pⁿ)1_V ⁿ

∂φˆn

(a) Given µ_n = P

p V_pⁿ

(d) Assume not (1): ∃p_n ∈ P_n s.t. diam(V_pⁿ) ≥ ε.

diam(∂φ(x)) ≥ ε where x = lim_n p_n ∈ X. lim_n→∞ kσ − σ_nk_L^∞_(X₎ = 0.

(1)

Up to extraction, φˆ_n −−−→^k.^k∞ φ, so that (2)

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Proof of the convergence theorem: Lower bound

X X X

spt(µ ) = P ⊆ X

∇φ

µ σ = ∇φ_#µ

I Step 2’:

p∈P_n

σ(V_pⁿ)

H^d(V_pⁿ)1_V ⁿ

∂φˆn

(a) Given µ_n = P

p V_pⁿ

(d) Assume not (1): ∃p_n ∈ P_n s.t. diam(V_pⁿ) ≥ ε.

diam(∂φ(x)) ≥ ε where x = lim_n p_n ∈ X. (e) Moreover, ∇φ_#ρ = σ. By Caffarelli’s regularity

lim_n→∞ kσ − σ_nk_L^∞_(X₎ = 0.

(1)

Up to extraction, φˆ_n −−−→^k.^k∞ φ, so that (2)

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4. Numerical results

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Computing the discrete Monge-Amp`ere operator

U(Gφ_#µ_P ) = P

with MA[φ](p) := H^d(∂φ(p)).ˆ

I Goal: fast computation of MA[φ](p) and its 1st/2nd derivatives w.r.t φ

(65)

Computing the discrete Monge-Amp`ere operator

X

Definition: Given a function φ : P → R,

I We rely on the notion of Laguerre (or power) cell in computational geometry

Lag^φ_P (p) := {y ∈ R^d; ∀q ∈ P, φ(q) ≥ φ(p) + hq − p|yi}

U(Gφ_#µ_P ) = P

Lag^φ_P(p)

p

(66)

Computing the discrete Monge-Amp`ere operator

X

U(Gφ_#µ_P ) = P

Lag^φ_P(p)

p

(67)

Computing the discrete Monge-Amp`ere operator

Lag^φ_P(p) := {y; ∀q ∈ P, kq − yk² ≥ kp − yk²} X

−→ For φ(p) = kpk²/2, one gets the Voronoi cell:

U(Gφ_#µ_P ) = P

Lag^φ_P(p)

p

−→ Computation in time O(|P | log |P|) in 2D