Hyperplane Tessellation

November 17th 2020

 

Oberseminar of Christoph Thäle's working group

at the

Ruhr University Bochum, Germany

 

gilles.bonnet@rub.de 

Hyperplane Tessellation

Construction

\(\widehat{X}\)   ... hyperplane process.

The complement \(\mathbb{R}^d \setminus ( \cup_{H\in \widehat{X}} H )\)  is a collection of (open) polytopes.

Their closures are the cells of the corresponding hyperplane tessellation \(X\)

Text

Very brief history

Very brief history

Note in [Schneider-Weil]:

Today: Signal Processing

Hyperplane Tessellation

Flat processes

Common assumptions:

  1. Stationary                           ... Intensity \( \gamma = \frac{1}{\kappa_{d-k}} \mathbb{E} X ( \mathcal{F}_{\mathrm{B}^d} ) \)
  2. Stationary and Isotropic ... Intensity measure \( = \gamma \mu_k \),
  3. homogeneous intensity measure \( \lambda (r \cdot ) = r^\alpha \lambda(\cdot)\),
  4. intensity measure of the form \( t \lambda(\cdot)\), \(t>0\).

Theorem: Intensity of stationary flat process [SW.4.4.3]

Let \(X\) be a stationary \(k\)-flat process in \(\mathbb{R}^d\) with intensity \(\gamma\). Then

 

where \(\lambda_E\) is the \(k\)-dimensional Lebesgue measure in \(E\).

            \(\lambda\) is the Lebesgue measure in \(\mathbb{R}^d\).

\mathbb{E} \sum_{E\in \widehat{X}} \lambda_E = \gamma \lambda
\iff \mathbb{E} \mathcal{H}^{k} \left[\left(\cup_{F\in \widehat{X}} \right) \cap W \right] = \gamma \lambda (W)

\(\widehat{X}\) is a point process in the affine Grassmannian \(A(d,k)\)

Hyperplanes processes

\(\widehat{X}\) is a point process in the affine Grassmannian \(A(d,d-1)\)

H(u,\tau)

o

Common assumptions:

  1. Stationary                           ... intensity \( \gamma = \frac{1}{2} \mathbb{E} X ( \mathcal{F}_{\mathrm{B}^d} ) \)
                                                   ... directional distribution \(\textcolor{red}{\varphi}\)
                                                   ... intensity measure \(\Theta\) where
    \[\int_{A(d,d-1)} f \ \mathrm{d} \Theta = \gamma \int_{S^{d-1}} \int_{-\infty}^{\infty} f(\textcolor{blue}{H(u,\tau)}) \ \mathrm{d}\tau \ \textcolor{red}{\varphi(\mathrm{d} u)}, \]
    \[ H(u,\tau) := \{ x\in\mathbb{R}^d : \langle x , u \rangle = \tau \} ,\]       
  2. Stationary and Isotropic ... Intensity measure \( = \gamma \mu_{d-1} \).

Intersection processes

\( \widehat{X} \)   ... point process in the affine Grassmannian \(A(d,d-1)\)

\(\widehat{X}_m\) ... intersection process of order \(m\) of \(X\), \(m\in\{1,\ldots,d \}\).

\[ \widehat{X}_m := \{ H_1 \cap\cdots\cap H_m : H_i \in X \ , \ H_i \neq H_j   \ ,\ 1\leq i<j\leq m\} .  \]

Theorem: Intensity of intersections [SW.4.4.8]

Let \(\widehat{X}\) be a stationary hyperplane process in \(\mathbb{R}^d\) with intensity \(\gamma\) and directional distribution \(\varphi\).

Then \(\widehat{X}_m\) is stationary and isotropic of intensity

\[ \frac{\gamma^k}{m!} \int_{S^{d-1}} \cdots \int_{S^{d-1}} \nabla_k(u_1,\ldots, u_m) \varphi (\mathrm{d} u_1) \ldots \varphi (\mathrm{d} u_m)\]

Intersection processes, CLT's

\(\widehat{X}\)   ... point process in the affine Grassmannian \(A(2,1)\),

\(n_R := F_0^1(B_R) \) ... number of line intersecting the ball \(B_R\),

\(I_R := F_0^2(B_R) \) ... number of intersection points in \(B_R\),

\(L_R := F_1^1(B_R) \) ... sum of the lengths of segments in \(B_R\).

Theorem: CLT (planar, binomial) [Paroux '06]

Let \(\widehat{X}\) be a stationary and isotropic Poisson line process in \(\mathbb{R}^2\) and \(R>0\). As \(R\to\infty\),

 

 

 

\frac{1}{I_R^{3/4}} \left( \textcolor{#980000}{I_R} - \frac{n_R(n_R-1)}{4} \right) \xrightarrow{\mathrm{d}} N\left( 0 , \frac{64}{3\pi^2} - 2 \right)
\frac{\textcolor{#980000}{L_R} - (\pi/2) R n_R}{L_R^{3/4}} \xrightarrow{\mathrm{d}} N\left( 0 , \frac{16}{3\pi^2} - \frac{1}{2} \right).

Poisson intersection

\(\widehat{X}\)   ... Poisson hyperplane process

\(\widehat{X}_m\)  ... intersection process of order \(m\) of \(\widehat{X}\),

\(F_i^m(W) :=\!\frac{1}{m!} \sum V_i (H_1 \cap \cdots \cap H_m \cap W) \) ...Intrinsic volume of \(\widehat{X}_m\) in \(W\!.\)

The proof relies on the so called Hoeffding’s decomposition of \(U\)-statistics.

In the isotropic setting, precise asymptotics for the variance are obtained when \(K\) is a ball, and ellipse or a rectangle.

In general \(\mathbb{V}F_k(rK) \) is of order \(r^{2d-1}\).

as \(r\to\infty\).

Theorem: CLT (count&volume)[Heinrich+2 Schmidt'06][Heinrich'09]

Let \(\widehat{X}\) be a stationary Poisson hyperplane process and \(W\) a fixed convex body. Assume that \(i\in\{0,d-k\}\), then



\frac{F_i^k(r W) - \mathbb{E} F_i^k(r W) }{\sqrt{\mathbb{V}F_i^k(rW)}} \xrightarrow{d} N(0,1),

Intersection processes

A different parametrization of the hyperplanes:

Intersections:

Result generalized to other expending window \(\rho K\) instead of \(B_r\).

Poisson intersection

Schulte and Reitzner show first a much more general CLT for \(U\)-statistics of Poisson processes. The proof relies on Malliavin calculus and the Wiener–Itô chaos expansion of \(U\)-statistics.

Theorem: CLT (all \(V_i\)'s + Wasserstein) [Schulte, Reitzner '13]

Let \(\widehat{X}\) be a Poisson hyperplane process of intensity measure \(\lambda \theta\), \(\lambda >0\) and \(W\) a fixed convex body. Assume that \(i\in\{0,\ldots,d-m\}\), then

 

d_W \left( \frac{F_i^m(W) - \mathbb{E} F_i^m(W) }{\sqrt{\mathbb{V}F_i^m(W)}} , N(0,1) \right) \leq c \, \lambda^{-\frac{1}{2}} .

Recall:  \(d_W(Y,Z) := \sup_{h\in\mathrm{Lip}(1)} |\mathbb{E}h(Y) - \mathbb{E}h(Z)|\).

\(\widehat{X}\)   ... Poisson hyperplane process

\(\widehat{X}_m\)  ... intersection process of order \(m\) of \(\widehat{X}\),

\(F_i^m(W) :=\!\frac{1}{m!} \sum V_i (H_1 \cap \cdots \cap H_m \cap W) \) ...Intrinsic volume of \(\widehat{X}_m\) in \(W\!.\)

Poisson intersection

\(\widehat{X}\)   ... Poisson flat process in \(A(d,k)\)

\(\widehat{X}_m\)  ... intersection process of order \(m\) of \(\widehat{X}\),

\(\psi \colon \mathcal{C}_o^d \subset \mathcal{C}^d \to \mathbb{R} \) satisfying mild assumptions,

\(F_\psi^m(W) :=\!\frac{1}{m!} \sum \psi (H_1 \cap \cdots \cap H_m \cap W) \) ... \(\psi\)-content of \(\widehat{X}_m\) in \(W\!,\)

\(\hat{F}_\psi^m (W) := t^{-m-\frac{1}{2}}(F_\psi^m(W)- \mathbb{E}F_\psi^m(W))\)

Theorem: Multivariate CLT [Last, Penrose, Schulte, Thäle '14]

1. Under reasonable assumption, one has

 

where \(\xi\) is a Gaussian field whose covariance function can be written as a geometric integral.

2. If \(\psi(r \, \cdot) = r^\alpha \psi(\cdot)\) then for \(t=1\),

 

d_3\bigl( (\hat{F}_\psi^m (W_1),\ldots,\hat{F}_\psi^m (W_\ell)) , (\xi (W_1),\ldots,\xi(W_\ell)) \bigr) \leq c_{(W_1,\ldots,W_\ell)} t^{-\frac{1}{2}}
d_3\bigl( (\hat{F}_\psi^m (\textcolor{#980000}{r} W_1),\ldots,\hat{F}_\psi^m (\textcolor{#980000}{r} W_\ell)) , (\xi (W_1),\ldots,\xi(W_\ell)) \bigr) \leq c_{(W_1,\ldots,W_\ell)} r^{-\frac{d-k}{2}}

Non Euclidean Setting ?

Hyperbolic 

\(\widehat{X}\)   ... Poisson process of hyperplanes (totally geodesic subspaces)

\(\widehat{X}_m\)  ... intersection process of order \(m\) of \(\widehat{X}\),

\(F^m(W) :=\!\frac{1}{m!} \sum \mathcal{H}^{d-m} (H_1 \cap \cdots \cap H_m \cap W) \)...Hausdorff measure of \(X_m\)

Methods:

Increase the intensity

Fix the window

 

CLT for any \(d\geq 2\) and any \(m\)

 

Increase the window

Fix the intensity

CLT for \(d\in\{2,3\}\) and any \(m\)

No CLT for \(d\geq 4\) and \(m=1\)

No CLT for \(d\geq 7\) and any \(m\)

\(\nLeftrightarrow\)

Spherical

\(\widehat{X}_m\)... intersection process of Poisson/Binomial great subspheres

\(W\) ... window = spherical cap

\(F^m(W)\) ... analogous as before

Question:

  1. Show CLT for fixed window
  2. Compute the variance as a function of the intensity and the radius of the cap. A different behavior between Poisson and Binomial case is expected.

Literature: I'm not sure about what is already know...

Hyperplane Tessellation

Tessellation (a.k.a. mosaic)

\(\widehat{X}\)   ... Poisson hyperplane process.

The complement \(\mathbb{R}^d \setminus ( \cup_{H\in \widehat{X}} H )\)  is a collection of (open) polytopes.

Their closures are the cells of the corresponding hyperplane tessellation \(X\)

For us, Tessellation = Marked Point Process where

  • Points = centers of the cells
    • center of mass
    • center of the inscribed/circumscribed ball
    • left most point
    • ...
  • Marks = Cells

If one chooses the most left point, the collection of centers is \(\widehat{X}_d\).

\(\Rightarrow\) the number of cells with center in \(W\) is \(F_0^d(W) \).

For a general notion of "tipicality" one needs to use Palm measure which definition can be found in:

Kallenberg (2007) "Random measure"

Cell intensity

\(X\)   ... stationary Poisson hyperplane tessellation

\(\gamma^{(d)}\) ... cell intensity := the intensity of the point process formed by the centers of all cells.

Your favorite book:

Tessellation (a.k.a. mosaic)

\(X\)   ... Poisson hyperplane tessellation

  1. Zero cell \( Z_o \)... cell containing the origin (always defined)                             
  2. Typical cell \(Z\) ... (require stationarity)
    • mark at a uniformly chosen center in a fixed window,
    • \(\Leftrightarrow\) volume-debiased version of \(Z_o\): \( \mathbb{E}f(Z_o) = \gamma^{(d)} \mathbb{E} [f(Z) V_d(Z)] \).
  • Relation \(Z_o\) <-> Typical Vornoi (Maybe)
  • Isotropy give large typical cell
  • Theorem 4.3 of O'Reilly (2020) (remark that there is a similar statement for the zero cell of Voronoi)
  • ...

One bit compression

\(x\in\mathbb{R}^d\!\!\) ... typical signal from a data set (\(\mathbb{R}^d\) or a Poisson process in \(\mathbb{R}^d\))

\((H(u_i,t_i))_{i\in I}\) ... stationary and isotropic hyperplane process

\(y_i = \mathrm{sign}(\langle u_i,x \rangle - t_i) \) ... measurements in \(\{-1,1\}\)

Quality of the compression:

  1. What is the distance to the closest data point \(y\) with a different encoding than \(x\) ?
  2. What is the distance to the farthest data point \(y\) with the same encoding than \(x\) ?
  3. If \(y\) is a displacement (Gaussian or fixed distance) of \(x\), what is the probability that \(x\) and \(y\) have the same encoding ?
  4. How much from the data set has the same encoding as \(x\) ?

Set \(x=o\) (Palm distribution)

\(x = o \in\mathbb{R}^d\) ... typical signal from a data set (\(\mathbb{R}^d\) or a Poisson process in \(\mathbb{R}^d\))

\((H(u_i,t_i))_{i\in I}\) ... stationary and isotropic hyperplane process

\(Z_o\)... the zero cell is the set of all points in  \(\mathbb{R}^d\) with the same enconding as \(o\).

Quality of the compression:

  1. What is the distance to the closest data point \(y\) outside of \(Z_o\)?
  2. What is the distance to the farthest data point \(y\) inside \(Z_0\) ?
  3. If \(y\) is a displacement (Gaussian or fixed distance) of \(o\), what is the probability that \(y\) is inside \(Z_o\) ?
  4. How much from the data set belong to \(Z_o\) ?

1. Closest point with a different encoding

\(x=o\in\mathbb{R}^d\!\!\) ... typical signal from a data set (in this slide: \(\mathbb{R}^d\)),

\((H(u_i,t_i))_{i\in I}\) ... Poisson hyperplane process: stationary, isotropic, intensity \(\gamma_d\).

The norm of the closest point to \(o\) in \(\mathbb{R}^d \setminus Z_o\) is \(r_\textrm{in}(Z_o)\),

where \(r_\textrm{in}(P) := \sup \{ r : B(o,r) \subset P \}\) for any polytope \(P\) containing the origin.

Proposition: Inradii of \(Z_o\) and \(Z\) [Baccelli, O'Reilly '19]

Assume that \(\gamma_d\to\infty\), then for any fixed \(r>0\)

   \(\lim_{d\to\infty} \mathbf{P}(r_{\mathrm{in}}(Z_o)>r) = 0\),

   \(\lim_{d\to\infty} \mathbf{P}(r_{\mathrm{in}}(Z)>r) = 0\).

Questions:

1) What if \(\gamma_d \to 0\) ? (super easy)

2) What if the data is a Poisson Process ?

2. Farthest point with a same encoding

\(x = o \in\mathbb{R}^d\!\!\) ... typical signal from a data set (in this slide: \(\mathbb{R}^d\)),

\((H(u_i,t_i))_{i\in I}\) ... Poisson hyperplane process: stationary, isotropic, intensity \(\gamma_d\).

The norm of the farthest point to \(x=0\) in \(\mathbb{R}^d \cap Z_o\) is \(r_\textrm{out}(Z_o)\),

where \(r_\textrm{out}(P) := \inf \{ r : P \subset B(o,r) \}\) for any polytope \(P\) containing the origin.

Theorem: Outradius of \(Z_o\) [Baccelli, O'Reilly '19]

Assume that \(\gamma_d \sim \rho d^\alpha\) and let \(R>0\).

Then there exist \(0<\rho_\ell < \frac{\sqrt{pi}}{R \sqrt{2}} < \rho_u\) such that

   \(\lim_{d\to\infty} \mathbf{P}(r_{\mathrm{out}}(Z_o)\leq d^{\frac{3}{2}-\alpha} R) = 0\), if \(\rho<\rho_\ell\),
   \(\lim_{d\to\infty} \mathbf{P}(r_{\mathrm{out}}(Z_o)\geq d^{\frac{3}{2}-\alpha} R) = 0\), if \(\rho>\rho_u\).

Method: Dualisation \(\to\) Poisson convex hull + Fine estimates

Questions:

1) Other regimes for \(\gamma_d\),

2) Typical cell ?

2. Farthest point with a same encoding

\(x = o \in\mathbb{R}^d\!\!\) ... typical signal from a data set,

\((H(u_i,t_i))_{i\in I}\) ... Poisson hyperplane process: stationary, isotropic, intensity \(\gamma_d\).

What if the data set is a stationary Poisson process \(\eta\) ?

 

This question makes sense only if \(Z_o\) contains points of the process.

 

We will come back to it in a few slides...

3. Displaced point

\(x = o \in\mathbb{R}^d\!\!\) ... typical signal from a data set (here: a stationary Poisson process \(\eta\) of intensity \(\lambda_d\)),

\((H(u_i,t_i))_{i\in I}\) ... Poisson hyperplane process: stationary, isotropic, intensity \(\gamma_d\),

Theorem: Separation of displaced point [Baccelli, O'Reilly '19]

Assume that \(\gamma_d \sim \rho d^\alpha\).

  1. Let \(G_{d,\sigma}\) be a \(\mathcal{N}(o,\sigma^2 I_d)\) Gaussian point in \(\mathbb{R}^d\). Then
    \[ \lim_{d\to\infty} \mathbf{P}(G_{d,\sigma}\in Z_o ) = \begin{cases} 0,& \alpha<0, \\ e^{-\sqrt{\frac{2}{\pi}}\rho\sigma}, & \alpha=0, \\ 1,& \alpha>0.\end{cases} \]
  2. Let \(U_{d,\delta}\) be a uniformly distributed point in \(\delta S^{d-1}\). Then
    \[ \lim_{d\to\infty} \mathbf{P}(U_{d,\delta}\in Z_o ) = \begin{cases} 0,& \alpha<\frac{1}{2}, \\ e^{-\sqrt{\frac{2}{\pi}}\rho\delta}, & \alpha=\frac{1}{2}, \\ 1,& \alpha>\frac{1}{2}.\end{cases} \]

4. Quantity of data with the same encoding

\(x = o \in\mathbb{R}^d\!\!\) ... typical signal from a data set (here: \(\mathbb{R}^d\)),

\((H(u_i,t_i))_{i\in I}\) ... Poisson hyperplane process: stationary, isotropic, intensity \(\gamma_d\).

Theorem: Expected volumes of \(Z_o\) and \(Z\)

\[ \mathbf{E} V_d(Z_o) = d! \kappa_d \left( \frac{d \kappa_d}{2 \gamma \kappa_{d-1}} \right)^d, \quad \mathbf{E} V_d(Z) = \frac{1}{\kappa_d} \left( \frac{d \kappa_d}{\gamma \kappa_{d-1}} \right)^d. \]

Corollary: Limits of \(\mathbf{E} Z_o\) and \(\mathbf{E} Z\) [Baccelli, O'Reilly '19]

Assume that \(\gamma_d = \rho d\), then

\[ \lim_{d\to\infty} \mathbf{E} V_d(Z_o) = \begin{cases} 0,& \rho<\frac{\pi}{\sqrt{e}}\\1,&\rho>\frac{\pi}{\sqrt{e}}\end{cases}, \quad \lim_{d\to\infty}\mathbf{E} V_d(Z) = \begin{cases} 0,& \rho<\frac{1}{\sqrt{e}}\\1,&\rho>\frac{1}{\sqrt{e}}.\end{cases} \]

4. Quantity of data with the same encoding

\(x = o \in\mathbb{R}^d\!\!\) ... typical signal from a data set (here: a stationary Poisson process \(\eta\) of intensity \(\lambda_d\))

\((H(u_i,t_i))_{i\in I}\) ... Poisson hyperplane process: stationary, isotropic, intensity \(\gamma_d\).

Theorem: Number of Poisson points in \(Z_o\) [Baccelli, O'Reilly '19]

Assume \( \lambda_d = d^{d(\alpha-1)} e^{d \lambda}\) and \(\gamma_d \sim \rho d^\alpha\), with \(\lambda \in\mathbb{R}\), \(\alpha\in \mathbb{R}\) and \(\rho>0\).

Set \(\rho^* := \frac{e^{\lambda}\pi}{\sqrt{e}}\). We have

\[ \lim_{d\to\infty} \mathbf{E} |(\eta\cap Z_o)\setminus\{o\}| = \begin{cases} 0,& \rho>\rho^*\\\infty,&\rho<\rho^* .\end{cases} \]

Thus, for \(\rho>\rho^*\),

\[ \lim_{d\to\infty} (\eta\cap Z_o = o) = 1 .\]

2. Farthest Poisson point with a same encoding

\(x = o \in\mathbb{R}^d\!\!\) ... typical signal from a data set (here: a stationary Poisson process \(\eta\) of intensity \(\lambda_d\))

Theorem: Farthest Poisson point in \(Z_o\) [Baccelli, O'Reilly '19]

Assume \( \lambda_d = d^{d(\alpha-1)} e^{d \lambda}\) with \(\lambda \in\mathbb{R}\), and \(\gamma_d \sim \rho d^\alpha\) with \(\rho < \rho^* := \frac{e^{\lambda}\pi}{\sqrt{e}}\).

  1. If \(\frac{\sqrt{\pi}}{r\sqrt{2}} < \rho^*\), then for all \(\rho\in(\frac{\sqrt{\pi}}{r\sqrt{2}},\rho^*)\),
    \[\limsup \frac{1}{d} \log \mathbb{P}\left( M_d \geq R d^{\frac{3}{2}-\alpha}  \right) \leq { \scriptsize \lambda + \frac{1}{2} \log2\pi e + \log R - \rho R \sqrt{\frac{2}{\pi}} + \log 4} .\]
  2. Thus, if ... conditions ..., then \(\lim \mathbb{P} \left( M_d \geq R d^{\frac{3}{2}-\alpha}  \right) = 0 .\)
  3. If \(\rho < \min(\frac{\sqrt{\pi}}{r\sqrt{2}},\rho^*)\),
    \[\limsup \frac{1}{d} \log \mathbb{P}\left( M_d \geq R d^{\frac{3}{2}-\alpha}  \right) \leq { \scriptsize \lambda + \frac{1}{2} \log2\pi e + \log R - \rho R \sqrt{\frac{2}{\pi}} + \log 4} .\]
  4. Thus, if \(R<\frac{1}{4e^{\lambda}\sqrt{2\pi}}\) and \(\rho< \min(\frac{\sqrt{\pi}}{r\sqrt{2}},\rho^*)\), then \( \lim \mathbb{P} \left( M_d \geq R d^{\frac{3}{2}-\alpha}  \right) = 0 .\)

\(M_d := \max \{\|x\| : x \in \eta \cap Z_o\}\).

Uniform point in \(Z_o\)

\lim_{d\to\infty} \mathbf{P}(\|Y_d\| \geq \sqrt{d} R ) = 0 \text{, if } R > R_u,
\limsup_{d\to\infty} \frac{1}{d} \ln \mathbf{P}(\|Y_d\|\geq \sqrt{d} R) \leq \tilde{c}(\rho,R) {\scriptsize \textcolor{gray}{:= \rho + \frac{1}{2}\ln\left(\frac{2e}{\pi}\right) + \ln R - e^\rho R \sqrt{\frac{2}{\pi e}},}} \text{ if } R > R_u .
\limsup_{d\to\infty} \frac{1}{d} \ln \mathbf{P}(\|Y_d\|\leq \sqrt{d} R) \leq c(\rho,R) {\scriptsize \textcolor{gray}{:= \rho + \frac{1}{2}\ln\left(\frac{\pi e}{2}\right) + \ln R - e^\rho R \sqrt{\frac{2}{\pi e}},}} \text{ if } R < R_u,

Theorem: Concentration in \(Z_o\) [O'Reilly'20]

Assumptions:

  • stationarity and isotropy,
  • cell intensity \(= e^{d \rho_d}\) with \(\rho_d \to \rho \in \mathbb{R}\) as \(d\to \infty\),
  • \(Y_n \sim \mathrm{Uniform}(Z_o - c(Z_o) )\), where \(c(Z_o)\) is the center of the inball.

Then, there exist \(0<R_\ell < R_u := e^{-\rho} \sqrt{\pi e/2}\) such that

 

 

 

 

\lim_{d\to\infty} \mathbf{P}(\|Y_d\| \leq \sqrt{d} R ) = 0 \text{, if } R < R_\ell,
  • Explicit representations of \(Z_o\) and \(Z\)
  • Carefull asymptotic analysis (Laplace method, etc )

Method:

Similar result for Poisson Voronoi.

  • Here the density of \(Y\) is log-concave
  • thus one can use the thin-shell phenomenon

Questions about hyperplanes

H1. For any of the model show concentration results

H2. Compute the variance in the spherical Binomial/Poisson setting

H... Suggestions ?

Questions about tessellation

T1. Concentration for the number of cells in a window \(\Rightarrow\) H1.

T... Suggestions ?