# Existence and approximation of densities of chord length- and cross section
area distributions

Thomas van der Jagt  Delft Institute of Applied Mathematics, Delft University
of Technology.  Geurt Jongbloed  Delft Institute of Applied Mathematics, Delft
University of Technology.  Martina Vittorietti  Scienze Economiche, Aziendali
e Statistiche, Università degli studi di Palermo.

###### Abstract

In various stereological problems an $n$-dimensional convex body is
intersected with an $(n-1)$-dimensional Isotropic Uniformly Random (IUR)
hyperplane. In this paper the cumulative distribution function associated with
the $(n-1)$-dimensional volume of such a random section is studied. This
distribution is also known as chord length distribution and cross section area
distribution in the planar and spatial case respectively. For various classes
of convex bodies it is shown that these distribution functions are absolutely
continuous with respect to Lebesgue measure. A Monte Carlo simulation scheme
is proposed for approximating the corresponding probability density functions.

##  1 Introduction

In a typical stereological problem we are presented with observations which
originate from a lower dimension than the dimension of interest. A classical
example is the Wicksell corpuscle problem [1]. The setting of the problem is
as follows, balls of varying size are randomly positioned in 3D space. This
system of balls is intersected with a plane and the circular section profiles
of the balls which happened to be cut by the section plane are observed. The
problem is to determine the distribution of the radii of the 3D balls given
the distribution of the radii of the observed 2D circular profiles.

An interesting generalization of this problem is to choose a convex shape
other than the ball for the shape of the particles. Then, the distribution of
observed section areas may be used to estimate the size distribution of the
particles. The particles we consider are convex bodies, i.e. compact and
convex sets with non-empty interior. In order to deal with such problems we
study a class of distributions which is especially important in this setting.
Suppose we take some convex body $K\subset\mathbb{R}^{3}$ of choice and
intersect $K$ with a random section plane. More generally, we may take a
convex body $K\subset\mathbb{R}^{n}$, and intersect it with a random
$(n-1)$-dimensional hyperplane. The random section planes we consider are
known as Isotropic Uniformly Random (IUR) planes. This roughly means that
every plane which has a non-empty intersection with the convex body has equal
probability of occurring. What can be said regarding the cumulative
distribution function (CDF) $G_{K}$ associated with the $(n-1)$-dimensional
volume of such a random section of $K$? In this paper we study this kind of
distribution functions. In particular, we obtain results on absolute
continuity. Whenever we refer to absolute continuity of a cumulative
distribution function we mean absolute continuity with respect to Lebesgue
measure. We know that the existence and the accurate approximation of the
density of $G_{K}$ is an essential ingredient for defining estimators for
particle size distributions in stereological problems. The existence results
and the approximation procedure proposed in this paper are used in [2] to
design a nonparametric maximum likelihood procedure for estimating particle
size distributions.

Given a convex body $K\subset\mathbb{R}^{2}$ an IUR section of $K$ is the
intersection of $K$ with a random line. The distribution function $G_{K}$ is
then also known as a chord-length distribution function. Some results
regarding this function may be found in [3]. The author notes that it is
typically assumed without proof that the CDF of a chord length distribution is
absolutely continuous. Only for a limited set of convex polygons there are
some results on absolute continuity. See for example [4] for the chord length
distribution function of a regular polygon which is absolutely continuous.

For convex bodies $K\subset\mathbb{R}^{3}$ the distribution function $G_{K}$
is sometimes called a cross section area distribution. In [5] it is noted that
in a stereological setting it is of interest to obtain the density of $G_{K}$
for some basic shapes such as the simplex or the cube. However, to the best of
our knowledge there are no results on whether $G_{K}$ has a density for a
large class of convex bodies, especially in $\mathbb{R}^{n}$ with $n\geq 3$.
To overcome the difficulty in obtaining an expression for $G_{K}$, simulations
may be used to find an approximation. In [6] a description is given for
approximating $G_{K}$ (when $K$ is a polytope in $\mathbb{R}^{3}$) and for how
it may be used to estimate the size distribution of particles from a sample of
observed section areas.

The problem we address is two-fold, first we show that for a large class of
convex bodies, $G_{K}$ is absolutely continuous. Second, we propose a Monte
Carlo simulation scheme to approximate the corresponding probability density
function using density estimation techniques. The outline of this paper is as
follows. In the second section necessary notation and definitions are
introduced. In the third section we discuss various results on the
distribution function $G_{K}$. Finally, in the fourth section we propose the
aforementioned Monte Carlo simulation scheme.

##  2 Preliminaries

In this section we introduce the necessary notation and definitions. In
particular, we introduce some terminology from convex geometry, a standard
reference is [7]. In $\mathbb{R}^{n}$ a convex body is a convex and compact
set with non-empty interior. Let $\mathcal{K}^{n}$ denote the class of convex
bodies in $\mathbb{R}^{n}$. Let $\operatorname{V}_{n}(K)$ be the
$n$-dimensional volume of $K$, its $n$-dimensional Lebesgue measure. $K$ and
$L$ indicate convex bodies. Given a point $x\in\mathbb{R}^{n}$, the
translation of $K$ with $x$ is given by: $K+x=\\{k+x:k\in K\\}$. The sum of
two sets, also known as the Minkowski sum, is defined as: $K+L=\\{k+l:k\in
K,l\in L\\}$. The dilatation or scaling of $K$ with $\lambda>0$ is given by:
$\lambda K=\\{\lambda k:k\in K\\}$. $\partial K$ denotes the boundary of $K$.
Given $x\in\mathbb{R}^{n}$ and $r>0$, we write
$B(x,r)=\\{y\in\mathbb{R}^{n}:||x-y||<r\\}$ and
$\bar{B}(x,r)=\\{y\in\mathbb{R}^{n}:||x-y||\leq r\\}$ for the open and closed
ball respectively, with radius $r$ centered at $x$. $\operatorname{SO}(n)$
denotes the rotation group of order $n$, containing all orthogonal $n\times
n$-matrices of determinant one. Given $M\in\operatorname{SO}(n)$, the rotation
of $K$ with $M$ is denoted by: $MK=\\{Mk:k\in K\\}$. We write
$\operatorname{int}K$ and $\operatorname{relint}K$ to denote the interior and
relative interior of $K$ respectively. A convex body $K\in\mathcal{K}^{n}$ is
strictly convex if for all $x,y\in K$ and $\lambda\in(0,1)$ we have $\lambda
x+(1-\lambda)y\in\operatorname{int}K$. A strictly convex body does not have
any line segments in its boundary. The unit sphere in $\mathbb{R}^{n}$ is
denoted by
$S^{n-1}=\\{(x_{1},\dots,x_{n})\in\mathbb{R}^{n}:x_{1}^{2}+\dots+x_{n}^{2}=1\\}$.
The upper hemisphere in $\mathbb{R}^{n}$ is given by:
$S_{+}^{n-1}=\\{(x_{1},\dots,x_{n})\in S^{n-1}:x_{n}\geq 0\\}$. Let
$\sigma_{n-1}$ denote the spherical measure on $S^{n-1}$, also known as the
spherical Lebesgue measure on $S^{n-1}$. In integrals over (a subset of)
$S^{n-1}$ the notation $\mathrm{d}\theta$ should be interpreted as
$\mathrm{d}\sigma_{n-1}(\theta)$. A hyperplane may be parameterized via a unit
normal vector $\theta\in S_{+}^{n-1}$ and its signed distance $s\in\mathbb{R}$
to the origin:

$T_{\theta,s}=\\{x\in\mathbb{R}^{n}:\langle x,\theta\rangle=s\\},$ (1)

with $\langle\cdot,\cdot\rangle$ being the usual inner product in
$\mathbb{R}^{n}$. Given a convex body $K\in\mathcal{K}^{n}$ its inner section
function $m_{K}:S^{n-1}\to[0,\infty)$ is defined by:

$m_{K}(\theta)=\max_{s\in\mathbb{R}}\operatorname{V}_{n-1}\left(K\cap
T_{\theta,s}\right).$ (2)

This function returns the maximal section volume for any given direction. We
can now define what is meant by an Isotropic, Uniformly Random (IUR) plane
hitting $K$. The following definition gives a convenient parameterization of
IUR planes, see [8] for IUR plane sections of convex bodies in
$\mathbb{R}^{3}$ (the generalization to $\mathbb{R}^{n}$ is straightforward):

######  Definition 1 (IUR plane).

An IUR plane $T$ hitting a fixed $K\in\mathcal{K}^{n}$, $n\geq 2$, is defined
as $T=T_{\Theta,S}$ where $(\Theta,S)$ has joint probability density,
$f_{K}:S_{+}^{n-1}\times\mathbb{R}\to[0,\infty)$ given by:

$f_{K}(\theta,s)=\begin{cases}\frac{1}{\mu([K])}&\text{if }K\cap
T_{\theta,s}\neq\emptyset\\\ 0&\text{otherwise,}\end{cases}$

with $T_{\theta,s}$ as in (1) and

$\mu([K])=\int_{S_{+}^{n-1}}\int_{-\infty}^{\infty}\mathds{1}{\\{K\cap
T_{\theta,s}\neq\emptyset\\}}\mathrm{d}s\mathrm{d}\theta.$ (3)

The notion of IUR planes was originally introduced in [9]. It is important to
highlight that there are other kinds of random planes which appear in
stereological problems, hence care should be taken in considering the
appropriate distribution. See [10] for more details. Note that the
distribution in Definition 1 is a joint uniform distribution; the marginals
are in general not uniform. We stress that the density $f_{K}$ prescribes the
probability associated with the possible locations and orientations of the
section plane, not the volumes of hyperplane sections. Fix
$K\in\mathcal{K}^{n}$ and let $f_{K}$ be as in Definition 1. Integrating out
the variable $s$, we obtain the marginal density:

$f_{K,\Theta}(\theta)=\frac{L(p_{\theta}(K))}{\mu([K])},\text{\quad}\theta\in
S_{+}^{n-1}.$ (4)

In Eq. (4), $p_{\theta}(K)$ represents the orthogonal projection of $K$ on the
line through the origin with direction $\theta$. $L(p_{\theta}(K))$ is then
the length of this orthogonal projection, hence $L(p_{\theta}(K))$ may also be
called the width of $K$ in direction $\theta$. The constant $\mu([K])$ is
related to the average width $\bar{b}(K)$, via:

$\mu([K])=\alpha_{n}\bar{b}(K).$ (5)

The average width is defined as:

$\bar{b}(K)=\frac{1}{\alpha_{n}}\int_{S_{+}^{n-1}}L(p_{\theta}(K))\mathrm{d}\theta,$

and the constant $\alpha_{n}$ is given by:

$\alpha_{n}=\sigma_{n-1}\left(S_{+}^{n-1}\right)=\frac{\pi^{\frac{n}{2}}}{\Gamma\left(\frac{n}{2}\right)}.$

Conditioning an IUR plane on a fixed direction yields a so-called Fixed
orientation Uniformly Random (FUR) plane. Fix $\theta\in S^{n-1}$, let
$a=a(\theta)$ be the smallest number such that $K\cap
T_{\theta,a}\neq\emptyset$, similarly let $b=b(\theta)$ be the largest number
such that $K\cap T_{\theta,b}\neq\emptyset$. Then, conditional on this
direction $\Theta=\theta$, $S$ is uniformly distributed on the interval
$[a,b]$, and we denote this conditional density by:

$f_{S|\Theta}(s|\theta)=\begin{cases}\frac{1}{b(\theta)-a(\theta)}&\text{if
}s\in[a(\theta),b(\theta)]\\\ 0&\text{otherwise,}\end{cases}$ (6)

We may also write: $S|\Theta=\theta\sim\mathcal{U}(a(\theta),b(\theta))$.
Given a cumulative distribution function (CDF) $F$ or a probability density
function (PDF) $f$, we write $X\sim F$ or $X\sim f$ to indicate that the
random variable $X$ is distributed according to $F$ or $f$ respectively. We
are now ready to introduce the CDF of interest in this paper.

######  Definition 2 (section volume CDF).

Fix $K\in\mathcal{K}^{n}$, $n\geq 2$, let $f_{K}$ be as in Definition 1. Let
$(\Theta,S)\sim f_{K}$, the random variable $Z=\operatorname{V}_{n-1}(K\cap
T_{\Theta,S})$ has cumulative distribution function $G_{K}$ which is given by:

$\displaystyle
G_{K}(z)=\int_{S_{+}^{n-1}}\int_{\mathbb{R}}\mathds{1}{\\{\operatorname{V}_{n-1}(K\cap
T_{\theta,s})\leq z\\}}f_{K}(\theta,s)\mathrm{d}s\mathrm{d}\theta.$

We refer to $G_{K}$ as the section volume CDF of $K$.

We remark that the expression of the CDF $G_{K}$ follows from the fact that
$G_{K}(z)=P(Z\leq z)=\mathbb{E}(\mathds{1}{\\{Z\leq z\\}})$ and the law of the
unconscious statistician. In $\mathbb{R}^{2}$ we may still refer to $G_{K}$ as
chord length distribution function and in $\mathbb{R}^{3}$ we may call it
cross section area distribution function.

##  3 Properties of the section volume CDF

In this section we derive various properties of the section volume CDF as
described in Definition 2. Given a convex body $K$ the following Lemma
highlights some basic properties.

######  Lemma 1.

Fix $K,L\in\mathcal{K}^{n}$, let $G_{K}$, $G_{L}$ be their section volume CDF
respectively. Let $z\in\mathbb{R}$, then:

  1. 1.

Translation invariance: $G_{K+x}(z)=G_{K}(z)$ for all $x\in\mathbb{R}^{n}$.

  2. 2.

Rotation invariance: $G_{MK}(z)=G_{K}(z)$ for all $M\in\operatorname{SO}(n)$.

  3. 3.

Scaling: $G_{\lambda K}(z)=G_{K}\left(z/\lambda^{n-1}\right)$ for all
$\lambda>0$.

  4. 4.

Inclusion: If $K\subset L$ then $G_{L}(z)\leq G_{K}(z)$.

The translation and rotation invariance of IUR planes is a defining property
of IUR planes, [9], and it may be used to prove property 1 and 2 in Lemma 1.
The third property also appears in [5] for $n=3$. All of these properties are
well-known for chord length distributions and the generalization to
$(n-1)$-dimensional sections of convex bodies in $\mathbb{R}^{n}$ is not
difficult. For the sake of completeness, the proof of this Lemma may be found
in the Appendix at the end of this paper.

We need Brunn’s theorem (see for example [11]) to prove one of our main
results:

######  Theorem 1 (Brunn).

Let $K\in\mathcal{K}^{n}$, $n\geq 2$. Fix $\theta\in S^{n-1}$. The function
$f_{\theta}:\mathbb{R}\to[0,\infty)$ given by:

$f_{\theta}(s)=\operatorname{V}_{n-1}(K\cap T_{\theta,s})^{\frac{1}{n-1}}$

is concave on its support.

Ignoring the exponent $1/(n-1)$ in the definition of $f_{\theta}$, this
function returns the volume of the intersection of $K$ with $T_{\theta,s}$.
Because we fix $\theta$ this means the function considers volumes of parallel
slices of $K$, and it is a function of the (signed) distance of the section
plane to the origin. The statement of Brunn’s Theorem inspires us to study a
distribution function which is closely related to $G_{K}$:

######  Definition 3 (Transformed section volume CDF).

Fix $K\in\mathcal{K}^{n}$, $n\geq 2$, let $f_{K}$ be as in Definition 1. Let
$(\Theta,S)\sim f_{K}$, the random variable $Z=\operatorname{V}_{n-1}(K\cap
T_{\Theta,S})^{1/(n-1)}$ has cumulative distribution function $G_{K}^{S}$
which is given by:

$\displaystyle G_{K}^{S}$
$\displaystyle(z)=\int_{S_{+}^{n-1}}\int_{\mathbb{R}}\mathds{1}{\\{\operatorname{V}_{n-1}(K\cap
T_{\theta,s})^{\frac{1}{n-1}}\leq
z\\}}f_{K}(\theta,s)\mathrm{d}s\mathrm{d}\theta.$

We refer to $G_{K}^{S}$ as the transformed section volume CDF of $K$.

This distribution function $G_{K}^{S}$ turns out to be more natural to study
compared to $G_{K}$. This will become clear in the proof of the upcoming
theorem. Note that, for $K\in\mathcal{K}^{n}$, $G_{K}$ and $G_{K}^{S}$ are
related as follows:

$G_{K}^{S}(z)=G_{K}\left(z^{n-1}\right).$

######  Remark 1.

$G_{K}$ is absolutely continuous if and only if $G_{K}^{S}$ is absolutely
continuous. After all, suppose that $G_{K}^{S}$ has probability density
function $g_{K}^{S}$. Let $X\sim g_{K}^{S}$, then $X^{n-1}\sim G_{K}$ and via
the well-known change of variables formula this random variable has the
following probability density function:

$g_{K}(z)=g_{K}^{S}\left(z^{\frac{1}{n-1}}\right)\frac{z^{\frac{2-n}{n-1}}}{n-1}.$
(7)

The converse case is analogous.

We now present one of the main theorems in this paper:

######  Theorem 2.

Let $K\in\mathcal{K}^{n}$, $n\geq 2$. Define the function
$f_{\theta}:\mathbb{R}\to[0,\infty)$ by:

$f_{\theta}(s)=\operatorname{V}_{n-1}(K\cap T_{\theta,s})^{\frac{1}{n-1}}.$

If $f_{\theta}$ has a unique maximum and is continuous on $\mathbb{R}$ for
almost all $\theta\in S_{+}^{n-1}$, then $G_{K}$ is absolutely continuous with
respect to Lebesgue measure.

###### Proof.

Given $K\in\mathcal{K}^{n}$, let $G_{K}$ be its section volume CDF and let
$G_{K}^{S}$ be its transformed section volume CDF. We show that $G_{K}^{S}$ is
absolutely continuous, from this it follows that $G_{K}$ is absolutely
continuous by Remark 1. By conditioning the distribution function $G_{K}^{S}$
on $\Theta$ having a particular value, $G_{K}^{S}$ may be written as a mixture
distribution:

$\displaystyle G_{K}^{S}(z)=\mathbb{P}\left(f_{\Theta}(S)\leq
z\right)=\int_{S_{+}^{n-1}}\mathbb{P}\left(f_{\theta}(S)\leq
z\Big{|}\Theta=\theta\right)f_{K,\Theta}(\theta)\mathrm{d}\theta,$

with $f_{K,\Theta}(\theta)$ being the marginal density of $\Theta$ as in (4)
and $f_{\theta}(\cdot)$ as in the statement of the theorem. For notation
convenience, write:

$G_{K}^{S}(z|\theta):=\mathbb{P}\left(f_{\theta}(S)\leq
z\Big{|}\Theta=\theta\right).$ (8)

Let $a=a(\theta)$ and $b=b(\theta)$ be as in (6) such that
$S|\Theta=\theta\sim\mathcal{U}(a,b)$. Choose $\theta\in S_{+}^{n-1}$ such
that $f_{\theta}$ has a unique maximum and is continuous on $\mathbb{R}$. By
definition of $a$ we know that $T_{\theta,a}$ intersects $K$ only through the
boundary of $K$. By the assumed continuity of $f_{\theta}$ we have
$f_{\theta}(a)=0$ and similarly: $f_{\theta}(b)=0$. Note that $f_{\theta}$ has
the following domain and codomain:

$f_{\theta}:[a,b]\to D_{\theta},\text{\ with \
}D_{\theta}=\left[0,m_{K}(\theta)^{\frac{1}{n-1}}\right],$ (9)

and $m_{K}(\cdot)$ as in (2). By Brunn’s theorem $f_{\theta}$ is concave on
its support and by assumption it attains its maximum in a single point $c$. As
a result, $f_{\theta}$ is strictly increasing on $(a,c)$ and strictly
decreasing on $(c,b)$. Therefore, $f_{\theta}$ restricted to $(a,c)$ is
invertible, and its inverse is convex and strictly increasing. Let:

$f_{\theta}^{+}:\left(0,m_{K}(\theta)^{\frac{1}{n-1}}\right)\to(a,c),$

denote this inverse. Similarly, $f_{\theta}$ restricted to $(c,b)$ has an
inverse:

$f_{\theta}^{-}:\left(0,m_{K}(\theta)^{\frac{1}{n-1}}\right)\to(c,b),$

which is concave and strictly decreasing. Write:

$p:=\mathbb{P}(S\in(a,c)|\Theta=\theta).$

By using the fact that $S|\Theta=\theta\sim\mathcal{U}(a,b)$ we find
$p=(c-a)/(b-a)$. Moreover, we obtain the following expression for
$G_{K}^{S}(z|\theta)$:

$\displaystyle G_{K}^{S}(z|\theta)$
$\displaystyle=\mathbb{P}\left(f_{\theta}(S)\leq
z\Big{|}\Theta=\theta,S\in(a,c)\right)p+\mathbb{P}\left(f_{\theta}(S)\leq
z\Big{|}\Theta=\theta,S\in(c,b)\right)(1-p)$
$\displaystyle=\mathbb{P}\left(S\leq
f_{\theta}^{+}(z)\Big{|}\Theta=\theta,S\in(a,c)\right)p+\mathbb{P}\left(S\geq
f_{\theta}^{-}(z)\Big{|}\Theta=\theta,S\in(c,b)\right)(1-p)$
$\displaystyle=\frac{f_{\theta}^{+}(z)-a}{c-a}p+\left(1-\frac{f_{\theta}^{-}(z)-c}{b-c}\right)(1-p).$

Because $f_{\theta}^{-}$ is concave and strictly decreasing, $-f_{\theta}^{-}$
is convex and strictly increasing. Therefore, $G_{K}^{S}(\cdot|\theta)$ is a
convex combination of two functions both of which are convex and strictly
increasing on the interval $D_{\theta}$ (as in Eq. (9)). As a result,
$G_{K}^{S}(\cdot|\theta)$ is convex, continuous, and strictly increasing on
$D_{\theta}$, which is the support of this distribution function. We conclude
that for almost all $\theta\in S_{+}^{n-1}$, $G_{K}^{S}(\cdot|\theta)$ is
absolutely continuous because it is convex on its support and continuous on
$\mathbb{R}$. Finally, this means that $G_{K}^{S}$ as a mixture of absolutely
continuous distribution functions is absolutely continuous by Fubini’s
theorem. ∎

######  Remark 2.

The arguments used in the proof of Theorem 1 do not hold for general convex
bodies. For general convex bodies the function $f_{\theta}$ is concave by
Brunn’s theorem. Therefore, the set of points at which it attains its maximum
may be an interval rather than a single point. When this is the case,
$G_{K}^{S}(\cdot|\theta)$ is still convex on its support, but it is
discontinuous in the point $m_{K}(\theta)^{1/(n-1)}$, which is the right
boundary point of its support. As a result, for any convex body
$K\in\mathcal{K}^{n}$, $G_{K}^{S}$ is convex on the interval $(0,d_{K})$ with:
$d_{K}=\min_{\theta\in S^{n-1}}m_{K}(\theta)^{1/(n-1)}$.

###  3.1 Strictly convex bodies

Let us now consider a particular class of convex bodies known as strictly
convex bodies. The class of strictly convex bodies is large in a precise
sense. For one, the class of convex bodies which are not smooth or strictly
convex form a set of first Baire category, see [12] for details. We have not
yet mentioned smooth convex bodies, loosely speaking it means that their
boundary is smooth. An important result we obtain in this section is that
given that $K\in\mathcal{K}^{n}$ is strictly convex, then $G_{K}$ is
absolutely continuous. Therefore, we show that for a large class of convex
bodies $G_{K}$ is absolutely continuous. The main tools to obtain this result
are the famous Brunn-Minkowski inequality and a variant of Brunn’s theorem. In
the field of convex geometry the importance of the Brunn-Minkowski inequality
cannot be overstated, we refer to the review paper [13] for variants of the
theorem and its applications.

######  Theorem 3 (Brunn-Minkowski).

Given convex bodies $K,L\in\mathcal{K}^{n}$ and $0<\lambda<1$ the following
inequality holds:

$\operatorname{V}_{n}(\lambda
K+(1-\lambda)L)^{\frac{1}{n}}\geq\lambda\operatorname{V}_{n}(K)^{\frac{1}{n}}+(1-\lambda)\operatorname{V}_{n}(L)^{\frac{1}{n}},$

with equality if and only if $K$ and $L$ are equal up to translation and
dilatation.

The equality condition in Theorem 3 means that there exist $\delta>0$ and
$x\in\mathbb{R}^{n}$ such that $K=\delta L+x$. In order to prove that $G_{K}$
is absolutely continuous for strictly convex $K\in\mathcal{K}^{n}$ we show
that the conditions in Theorem 2 are satisfied. First, we need the following
Lemma:

######  Lemma 2.

Let $K,L\in\mathcal{K}^{n}$ with $K\subset\operatorname{int}L$, then
$\operatorname{V}_{n}(K)<\operatorname{V}_{n}(L)$.

Its proof is given in the Appendix at the end of this paper. We show that the
strict convexity of a convex body carries over to strict concavity of the
function $f_{\theta}$ (as in Theorem 1).

######  Theorem 4.

Let $K\in\mathcal{K}^{n}$ be a strictly convex body, $n\geq 2$. Fix $\theta\in
S^{n-1}$. The function $f_{\theta}:\mathbb{R}\to[0,\infty)$ given by:

$f_{\theta}(s)=\operatorname{V}_{n-1}(K\cap T_{\theta,s})^{\frac{1}{n-1}}$

is continuous on $\mathbb{R}$ and strictly concave on its support.

###### Proof.

The proof is a slight variation of a proof of Brunn’s theorem using the Brunn-
Minkowski inequality as found in [11] (pp 18, 19). Fix $\theta\in S^{n-1}$.
Choose $r,t$ in the support of $f_{\theta}$, such that $r<t$. Let
$\lambda\in(0,1)$, set $s=\lambda r+(1-\lambda)t$ and consider the hyperplane
sections $K_{r}:=K\cap T_{\theta,r}$, $K_{s}:=K\cap T_{\theta,s}$ and
$K_{t}:=K\cap T_{\theta,t}$. We show that:

$\lambda K_{r}+(1-\lambda)K_{t}\subset(\operatorname{int}K)\cap T_{\theta,s}.$
(10)

Let $z\in\lambda K_{r}+(1-\lambda)K_{t}$, then $z=\lambda x+(1-\lambda)y$ for
some $x\in K_{r}$ and some $y\in K_{t}$. Because also $x,y\in K$ we have
$z\in\operatorname{int}K$ due to the strict convexity of $K$. Also, note that
$\langle z,\theta\rangle=\lambda\langle x,\theta\rangle+(1-\lambda)\langle
y,\theta\rangle=\lambda r+(1-\lambda)t=s$. Hence, $z\in T_{\theta,s}$, which
proves (10). It can readily be verified that: $(\operatorname{int}K)\cap
T_{\theta,s}=\operatorname{relint}\left(K_{s}\right)$. Combining this with
(10) we find: $\lambda
K_{r}+(1-\lambda)K_{t}\subset\operatorname{relint}\left(K_{s}\right)$. Let
$\Pi(L)$ denote the orthogonal projection of $L$ on the hyperplane
$T_{\theta,0}$. Note that $\lambda K_{r}+(1-\lambda)K_{t}$ and $K_{s}$ are
subsets of $T_{\theta,s}$, projecting them on $T_{\theta,0}$ preserves the
inclusion:

$\Pi(\lambda K_{r}+(1-\lambda)K_{t})\subset\operatorname{relint}\Pi(K_{s}).$

Identifying $T_{\theta,0}$ with $\mathbb{R}^{n-1}$ we may regard $\Pi(\lambda
K_{r}+(1-\lambda)K_{t})$ and $\Pi(K_{s})$ as convex bodies in
$\mathbb{R}^{n-1}$. Under this identification $\Pi(\lambda
K_{r}+(1-\lambda)K_{t})\subset\operatorname{int}\Pi(K_{s})$ and applying Lemma
2 yields:

$\operatorname{V}_{n-1}\left(\Pi(\lambda
K_{r}+(1-\lambda)K_{t})\right)<\operatorname{V}_{n-1}\left(\Pi(K_{s})\right).$

Keep in mind that projecting a set on $T_{\theta,0}$ does not affect its
$(n-1)$-dimensional volume. Note that we may change the order of these
projections and the Minkowski sum:

$\lambda\Pi(K_{r})+(1-\lambda)\Pi(K_{t})=\Pi(\lambda K_{r}+(1-\lambda)K_{t}),$

where the sum of sets is considered in the plane $T_{\theta,0}$. Hence,

$\operatorname{V}_{n-1}\left(\Pi(K_{s})\right)>\operatorname{V}_{n-1}(\Pi(\lambda
K_{r}+(1-\lambda)K_{t}))=\operatorname{V}_{n-1}(\lambda\Pi(K_{r})+(1-\lambda)\Pi(K_{t})).$
(11)

Once again, $\Pi(K_{r})$ and $\Pi(K_{t})$ may be identified as convex bodies
in $\mathbb{R}^{n-1}$ and we may apply Brunn-Minkowski’s (B.M.) inequality to
obtain the desired result:

$\displaystyle f_{\theta}(s)$
$\displaystyle=\operatorname{V}_{n-1}(K_{s})^{\frac{1}{n-1}}$
$\displaystyle=\operatorname{V}_{n-1}(\Pi(K_{s}))^{\frac{1}{n-1}}$
$\displaystyle{>}\operatorname{V}_{n-1}(\lambda\Pi(K_{r})+(1-\lambda)\Pi(K_{t}))^{\frac{1}{n-1}}$
$\displaystyle{\geq}\lambda\operatorname{V}_{n-1}(\Pi(K_{r}))^{\frac{1}{n-1}}+(1-\lambda)\operatorname{V}_{n-1}(\Pi(K_{t}))^{\frac{1}{n-1}}$
$\displaystyle=\lambda\operatorname{V}_{n-1}(K_{r})^{\frac{1}{n-1}}+(1-\lambda)\operatorname{V}_{n-1}(K_{t})^{\frac{1}{n-1}}$
$\displaystyle=\lambda f_{\theta}(r)+(1-\lambda)f_{\theta}(t).$

Continuity of $f_{\theta}(\cdot)$ can be shown as follows. Let $a=a(\theta)$
and $b=b(\theta)$ be as in (6). By definition of $a$ we know that
$T_{\theta,a}$ intersects $K$ only through the boundary of $K$. This
intersection only contains a single point, if another point were in the
intersection this would imply that the boundary of $K$ contains a line segment
which contradicts the strict convexity of $K$. As a result: $f_{\theta}(a)=0$
and similarly: $f_{\theta}(b)=0$. Because $a$ and $b$ are the only possible
points of discontinuity, $f_{\theta}(\cdot)$ is continuous. ∎

Because a bounded concave function has a maximum, strict concavity then
implies that the maximum is unique. We obtain as a direct consequence of
Theorem 1 and Theorem 2:

######  Corollary 1.

Let $K\in\mathcal{K}^{n}$ strictly convex, and let $G_{K}$ be its section
volume CDF. Then, $G_{K}$ is absolutely continuous.

###  3.2 Non-strictly convex bodies

In this section we obtain some results for convex bodies which are not
necessarily strictly convex. We show that for any $K\in\mathcal{K}^{n}$ the
CDF $G_{K}$ can be approximated arbitrarily closely by the CDF $G_{L}$ for
some strictly convex $L\in\mathcal{K}^{n}$. This is due to the fact that any
convex body may be approximated by a smooth and strictly convex body. A
quantitatively useful statement is the following, see Theorem 1.5 in [14]:

######  Lemma 3.

Let $K\in\mathcal{K}^{n}$ a convex body with $0\in\operatorname{int}K$. Let
$0<\lambda<1$. There exists a smooth and strictly convex body
$L\in\mathcal{K}^{n}$ such that:

$\lambda K\subset L\subset K.$

######  Theorem 5.

Given a convex body $K\in\mathcal{K}^{n}$, there exists a sequence of strictly
convex bodies $K_{m}\in\mathcal{K}^{n}$, $m\in\mathbb{N}$, such that the
sequence of the corresponding section volume CDFs
$(G_{K_{m}})_{m\in\mathbb{N}}$ converges pointwise to $G_{K}$ as $m\to\infty$.

###### Proof.

Let $K\in\mathcal{K}^{n}$ and let $G_{K}$ be its section volume CDF. By
property 1 of Lemma 1 we may assume without loss of generality that
$0\in\operatorname{int}K$. Set $\lambda_{m}:=1-1/(m+1)$ for $m\in\mathbb{N}$.
Then $0<\lambda_{m}<1$ and $\lambda_{m}$ increases to 1 as $m\to\infty$. Using
Lemma 3, for $\lambda_{m}$ let $K_{m}$ be a smooth and strictly convex body
such that: $\lambda_{m}K\subset K_{m}\subset K$. Let $z\in\mathbb{R}$, by
property 4 and 1 of Lemma 1 we find:

$G_{K}(z)\leq G_{K_{m}}(z)\leq
G_{\lambda_{m}K}(z)=G_{K}\left(\frac{z}{\lambda_{m}^{n-1}}\right).$

As a result we obtain the following bound:

$|G_{K_{m}}(z)-G_{K}(z)|\leq\left|G_{K}\left(\frac{z}{\lambda_{m}^{n-1}}\right)-G_{K}(z)\right|.$
(12)

Note that $z/\lambda_{m}^{n-1}$ decreases towards $z$ as $m\to\infty$. Because
$G_{K}$ is a CDF it is right-continuous, therefore:

$\lim_{m\to\infty}G_{K}\left(\frac{z}{\lambda_{m}^{n-1}}\right)=G_{K}(z).$

By Eq. (12) this finishes the proof. ∎

###  3.3 Polytopes

In this section we study polytopes, which are especially of interest for
practical applications. Being examples of non-strictly convex bodies, they are
not covered by Corollary 1. Via the following theorem we can prove that the
section volume CDF for a large class of polytopes is absolutely continuous.

######  Theorem 6.

Let $P\subset\mathbb{R}^{n}$ be a full-dimensional polytope, $n\geq 2$, such
that none of its edge-directions are parallel. Define the function
$f_{\theta}:\mathbb{R}\to[0,\infty)$ by:

$f_{\theta}(s)=\operatorname{V}_{n-1}(P\cap T_{\theta,s})^{\frac{1}{n-1}}.$

For almost all $\theta\in S_{+}^{n-1}$, $f_{\theta}$ is continuous on
$\mathbb{R}$ and has a unique maximum.

###### Proof.

Let $\theta\in S_{+}^{n-1}$. Let $a=a(\theta)$ and $b=b(\theta)$ as in (6).
For $s\in(a,b)$ the intersection $P\cap T_{\theta,s}$ is an
$(n-1)$-dimensional polytope, and its vertices are the intersections of
$T_{\theta,s}$ with the edges of $P$. For $s\in(a,b)$ write:

$L(s)=P\cap T_{\theta,s},$

for these polytopes obtained by intersections. By Brunn’s theorem we know that
$f_{\theta}$ is concave on its support. The set of points at which a concave
function attains its maximum is convex, hence it is an interval:
$[s_{-},s_{+}]$, or a single point. Assume that $s_{-}<s_{+}$, we show that
this leads to a contradiction. By Brunn-Minkowski’s inequality, and in
particular its equality condition, we know that all sections
$\\{L(s):s\in[s_{-},s_{+}]\\}$ are equal up to dilatation and translation.
But, because all such sections have equal volume, these sections then have to
be equal up to translations. The interval $[a,b]$ may be divided in a finite
number of disjoint intervals $[\alpha_{i},\beta_{i}]$, $i\in\\{1,2,\dots,m\\}$
such that for each $s$ in such an interval the plane $T_{\theta,s}$ intersects
the same set of edges. Let $e_{1},e_{2}\subset P$ be two edges which are
intersected by all section planes $T_{\theta,s}$ with
$s\in[\alpha_{i},\beta_{i}]$ for some $i$. Because $e_{1}$ and $e_{2}$ are not
parallel, the distance function:

$s\mapsto||T_{\theta,s}\cap e_{1}-T_{\theta,s}\cap e_{2}||,$

is a continuous function on $[\alpha_{i},\beta_{i}]$, which is strictly
increasing or strictly decreasing. This means that if we compare the resulting
polytopes $\\{L(s):s\in[\alpha_{i},\beta_{i}]\\}$, the distances between any
of their vertices change continuously as a function of $s$. Therefore, the
sections $P\cap T_{\theta,s}$ for $s\in[\alpha_{i},\beta_{i}]$ cannot all be
equal up to a translation. This holds for any interval
$[\alpha_{i},\beta_{i}]$ which is a contradiction.

Now it remains to show that $f_{\theta}$ is continuous on $\mathbb{R}$ for
almost all $\theta\in S_{+}^{n-1}$. Note that for almost all $\theta\in
S_{+}^{n-1}$ the section planes $T_{\theta,s}$ enter the polytope through a
vertex and leave the polytope through a vertex as $s$ runs from $a(\theta)$ to
$b(\theta)$. For any such $\theta$, $f_{\theta}(a)=0$ and $f_{\theta}(b)=0$,
because a vertex has no $(n-1)$-dimensional volume. As $a$ and $b$ are the
only possible points of discontinuity it follows that $f_{\theta}$ is
continuous on $\mathbb{R}$ for almost all $\theta\in S_{+}^{n-1}$. ∎

######  Example 1 (Cube).

Let $\alpha\geq 0$ and define
$P_{\alpha}=\\{x\in\mathbb{R}^{3}:A_{\alpha}x\leq b\\}$, with $A_{\alpha}$ and
$b$ given by:

$A_{\alpha}=\begin{pmatrix}0&\alpha&1\\\ \alpha&0&-1\\\ 1&0&\alpha\\\
-1&\alpha&0\\\ \alpha&1&0\\\
0&-1&\alpha\end{pmatrix}b=\begin{pmatrix}\frac{1}{2}\\\ \frac{1}{2}\\\
\frac{1}{2}\\\ \frac{1}{2}\\\ \frac{1}{2}\\\ \frac{1}{2}\end{pmatrix}$

Note that for $\alpha=0$ we obtain the centered unit cube:
$P_{0}=[-\frac{1}{2},\frac{1}{2}]^{3}$. For a small $\alpha>0$ we obtain a
sort of perturbed cube, such that none of its edge-directions are parallel,
see Figure 1. We choose an arbitrary direction $\theta\in S_{+}^{2}$ and
visualize $f_{\theta}$ (as in Theorem 6) on its support for $P_{0}$ and
$P_{\alpha}$ with $\alpha=0.15$. The result is shown in Figure 1. We can
clearly see that for the cube $f_{\theta}$ is constant on some interval while
this is not the case for the perturbed cube.

As a direct consequence of Theorem 2 we obtain:

######  Corollary 2.

Let $P\subset\mathbb{R}^{n}$ be a full-dimensional polytope, $n\geq 2$, such
that none of its edge-directions are parallel. Let $G_{P}$ be its section
volume CDF. Then, $G_{P}$ is absolutely continuous.

######  Remark 3.

In $\mathbb{R}^{3}$ the condition in Theorem 6 may be weakened as follows.
Suppose we have a simple polytope $P\subset\mathbb{R}^{3}$, such that each
edge is parallel to at most one other edge. If we follow the steps of the
proof of Theorem 6, then all sections $P\cap T_{\theta,s}$ for
$s\in(s_{-},s_{+})$ intersect at least three edges of $P$, because $P$ is
simple. Therefore, these sections of the polytope also intersect two edges
which are not parallel. This is sufficient for reaching the contradiction in
the proof of Theorem 6. An example of a polytope satisfying this condition is
the convex dodecahedron, which has 30 edges, each edge is parallel to one
other edge.

Figure 1: Left: a perturbed version of the cube. Right: the function
$f_{\theta}$ visualized for a fixed $\theta\in S_{+}^{2}$, for the cube and
the perturbed version of the cube.

##  4 Density approximation in the absolutely continuous case

In this section we consider convex bodies $K\in\mathcal{K}^{n}$ such that
$G_{K}$ is absolutely continuous. For most convex bodies $K\in\mathcal{K}^{n}$
there is no known explicit expression for $G_{K}$ or its density $g_{K}$. In
this section we focus on approximating the density $g_{K}$. This is achieved
by obtaining a large sample from $G_{K}$ along with a kernel density estimator
(KDE). We use the following rejection sampling scheme, proposed in [10], to
sample from the distribution $G_{K}$:

  1. 1.

Enclose $K$ inside a sphere: choose $R>0$ such that $K\subset\bar{B}(0,R)$.

  2. 2.

Choose an isotropic random direction $\Theta\sim\mathcal{U}(S^{n-1})$.

  3. 3.

Sample $S\sim\mathcal{U}(0,R)$.

  4. 4.

The plane $T_{\Theta,S}$ hits $\bar{B}(0,R)$, if the plane also hits $K$ we
accept, and $Z=\operatorname{V}_{n-1}(K\cap T_{\Theta,S})$ is a draw from
$G_{K}$. If the plane does not hit $K$, we reject and go back to step 2.

In $\mathbb{R}^{2}$, step 2 may be achieved by sampling
$\Phi\sim\mathcal{U}(0,2\pi)$ followed by setting
$\Theta=(\cos(\Phi),\sin(\Phi))$. In $\mathbb{R}^{3}$ step 2 can be performed
as follows. Sample $\Phi\sim\mathcal{U}(0,2\pi)$ and $X\sim\mathcal{U}(-1,1)$.
Then, we may set $\Omega=\arccos\left(X\right)$ and
$\Theta=(\sin\Omega\cos\Phi,\sin\Omega\sin\Phi,\cos\Omega)$. In order to keep
the rejection rate in the sampling scheme low, $R$ should be as small as
possible and $K$ should be positioned at the origin, meaning that
$0\in\operatorname{int}K$.

Of course, for $K\in\mathcal{K}^{n}$ with $n=2$ we find: $G_{K}\equiv
G_{K}^{S}$. Note that $G_{K}^{S}$ is initially convex on some initial interval
(see Remark 2). As a result, its density $g_{K}^{S}$ is non-decreasing on this
interval. This means that $g_{K}^{S}$ may even be constant initially. In
addition, note that if $n=2$ and $K$ is a polygon then in [3] it has been
shown that $g_{K}^{S}$ is always constant on some initial interval. Because of
the relation between $g_{K}^{S}$ and $g_{K}$, if $g_{K}^{S}$ is constant on an
initial interval, then $g_{K}$ behaves like $z^{(2-n)/(n-1)}$ on this
interval. Hence, when $n=3$ this means that $g_{K}$ behaves like $1/\sqrt{z}$
for $z$ close to zero. Clearly, this complicates the approximation of $g_{K}$
near zero. Therefore, we choose to approximate the density $g_{K}^{S}$
instead, and use Eq. (7) to obtain an approximation of $g_{K}$.

We will now introduce the Monte Carlo simulation scheme for approximating
$g_{K}^{S}$. We choose a large $N\in\mathbb{N}$ and sample
$Z_{1},\dots,Z_{N}\overset{\mathrm{iid}}{\sim}G_{K}$ using the sampling scheme
given above. Setting $X_{i}=Z_{i}^{1/(n-1)}$, we obtain that
$X_{1},\dots,X_{N}\overset{\mathrm{iid}}{\sim}G_{K}^{S}$. The following KDE is
for example studied in [15], which we propose as an approximation for
$g_{K}^{S}$:

$\hat{g}_{N}^{S}(z)=\frac{1}{hN}\sum_{i=1}^{N}k\left(\frac{z-X_{i}}{h}\right)+k\left(\frac{z+X_{i}}{h}\right),\text{
\quad}z\geq 0,$ (13)

with $h>0$ the bandwidth parameter and $k$ a symmetric kernel. The KDE in (13)
is also known as the reflection method. A reason for using the reflection
method over the classical (Parzen-Rosenblatt) KDE is that it ensures that no
probability mass is assigned for $z<0$. Recall that the classical KDE for the
sample $X_{1},\dots,X_{N}$ is given by:

$f(z)=\frac{1}{hN}\sum_{i=1}^{n}k\left(\frac{z-X_{i}}{h}\right),\text{
\quad}z\in\mathbb{R}.$ (14)

Note that when computing the KDE in (14) for the following ’sample’ of size
$2N$:

$X_{1},X_{2},\dots,X_{N},-X_{1},-X_{2},\dots,-X_{N},$

we find: $f(z)=\hat{g}_{N}^{S}(z)/2$. This fact may be used to choose the
bandwidth $h$, since most of the literature is devoted to bandwidth selection
for the classical KDE. In the data examples in the next section we choose for
$k$ the Gaussian kernel and select the bandwidth with the popular Sheather-
Jones method [16]. Whenever we want to approximate $g_{K}$ instead of
$g_{K}^{S}$, we simply follow the procedure given above to compute
$\hat{g}_{N}^{S}$. Then, using Eq. (7) we set:

$\hat{g}_{N}(z)=\hat{g}_{N}^{S}\left(z^{\frac{1}{n-1}}\right)\frac{z^{\frac{2-n}{n-1}}}{n-1},$
(15)

which is an approximation of $g_{K}$. A drawback of the KDE in (13) is that
this density has (right)-derivative zero in $z=0$. As mentioned before, when
$n=2$ and $K$ is a convex polygon this is not an issue since the density
$g_{K}^{S}$ is then initially constant. In the data examples in the next
section the approximations of $G_{P}^{S}$ of some convex polytopes $P$ in
$\mathbb{R}^{3}$ appear initially (close to) linear. This suggests that the
choice of boundary correction is reasonable. Should one consider a polytope
$P$ such that $G_{P}^{S}$ is far from being initially linear then other
boundary correction methods may be more appropriate.

Figure 2: Left: 100 IUR sections through $P$ the unit square in
$\mathbb{R}^{2}$. Right: comparison of the density $g_{P}$ to its
approximation $\hat{g}_{N}$.

Figure 3: Approximations of $g_{P}^{S}$ and $g_{P}$ for $P$ the unit cube
(upper panel) and for $P$ the dodecahedron with volume 1 (lower panel).

###  4.1 Simulations

In this section we perform a few simulations to show that the Monte Carlo
simulation scheme works well. For these simulations we focus on polytopes.
Throughout this section, let $P\subset\mathbb{R}^{n}$ be a full-dimensional
polytope. We have implemented the sampling scheme for drawing samples from
$G_{P}$ specifically for $n=2$ and $n=3$. The code used for the simulations
may be found at https://github.com/thomasvdj/pysizeunfolder. The polytope can
be entered into this program either by presenting a set of points, such that
the polytope is given by the convex hull of these points, or by presenting a
half-space representation of the polytope.

In the literature, similar simulations have been performed, e.g. for the cube
and the dodecahedron. Therefore we also consider these shapes, such that we
have a point of comparison. We do realize that we have established absolute
continuity of $G_{P}^{S}$ and $G_{P}$ for a specific class of polytopes, and
the cube is not covered by Corollary 2. It is of course still possible to
follow the approximation procedure described in the previous section. We also
note that when we considered the perturbed version of the cube as in Example
1, for a small $\alpha>0$, the results were indistinguishable to the eye
compared to the results corresponding to an actual cube. By Remark 3 the
section volume CDF of the dodecahedron is absolutely continuous. For
approximating $G_{P}$ no assumptions on absolute continuity are required, as
it may be approximated arbitrarily closely by an empirical distribution
function, given a large sample from $G_{P}$.

For all simulations, we set $N=10^{7}$. For the first example, we choose the
unit square in $\mathbb{R}^{2}$. The density of its chord length distribution
may be found in [17], it is given by:

$g_{P}(z)=\begin{cases}\frac{1}{2}&\text{if }0\leq z\leq 1\\\
\frac{1}{z^{2}\sqrt{z^{2}-1}}-\frac{1}{2}&\text{if
}1<z\leq\sqrt{2}\end{cases}.$

The approximation obtained via the proposed Monte Carlo scheme is shown in
Figure 2. Figure 2 also contains a visualization of 100 IUR sections through
the unit square. As can be seen in Figure 2, the approximation $\hat{g}_{N}$
is very close to the true probability density $g_{P}$.

Figure 4: Approximations of $G_{P}^{S}$ for $P$ the unit cube (Left) and for
$P$ the dodecahedron with volume 1 (Right).

We should stress that the proposed method is especially useful in the spatial
setting $n=3$. Naturally, whenever the analytical expression for $g_{P}$ is
available this is preferable. To the best of our knowledge, there are no known
expressions for $g_{P}$ of any polytope $P$ in $\mathbb{R}^{3}$. In the planar
case ($n=2$) the density $g_{P}$ is known for various polygons, for example
for rectangles [17], and regular polygons [4]. In Figure 3 the approximations
of $g_{P}$ and $g_{P}^{S}$ are shown for the cube and the dodecahedron, both
shapes scaled to have volume 1.

Similar simulations were performed in [18] for the cube and dodecahedron,
qualitatively the curves visualized there are close to the approximations of
$g_{P}$ shown in Figure 3. For the cube, one can easily see that for any
direction $\theta\in S_{+}^{n-1}$, there exists a section of area 1. By Remark
2, this means that $g_{P}^{S}$ is non-decreasing on $(0,1)$, which can also be
seen in Figure 3. Approximations of $G_{P}^{S}$ for the cube and dodecahedron
are shown in Figure 4. For these visualizations the same samples are used as
in Figure 3. As mentioned before, these approximations of $G_{P}^{S}$ appear
initially (close to) linear, justifying the choice of boundary correction in
the density approximation procedure.

##  5 Concluding remarks

In this paper we establish absolute continuity of the (transformed) section
volume CDF for various classes of convex bodies. Whether these distribution
functions are absolutely continuous for all convex bodies remains an open
problem. From a theoretical perspective we cover a large class of convex
bodies with the strictly convex bodies. From a practical point of view we
consider a large class of polytopes, and for these polytopes we provide a
Monte Carlo simulation scheme for approximating the density corresponding to
its (transformed) section volume CDF.

##  Appendix A Appendix: additional proofs

###### Proof of Lemma 1.

Let $x\in\mathbb{R}^{n}$, $\theta\in S_{+}^{n-1}$ and $s\in\mathbb{R}$. It can
be easily verified that the following holds:

$(K+x)\cap T_{\theta,s}=\left(K\cap T_{\theta,s-\langle
x,\theta\rangle}\right)+x.$ (16)

Meaning that the intersection of a translated $K$ with a plane is the same as
the intersection of $K$ with a translated plane and then translating the
result. It follows that:

$\displaystyle G_{K+x}(z)$
$\displaystyle=\int_{S_{+}^{n-1}}\int_{\mathbb{R}}\frac{\mathds{1}{\\{\operatorname{V}_{n-1}((K+x)\cap
T_{\theta,s})\leq z\\}}\mathds{1}{\\{(K+x)\cap
T_{\theta,s}\neq\emptyset\\}}}{\mu([K+x])}\mathrm{d}s\mathrm{d}\theta$
$\displaystyle{=}\int_{S_{+}^{n-1}}\int_{\mathbb{R}}\frac{\mathds{1}{\\{\operatorname{V}_{n-1}((K\cap
T_{\theta,s-\langle x,\theta\rangle})+x)\leq z\\}}\mathds{1}{\\{K\cap
T_{\theta,s-\langle
x,\theta\rangle}\neq\emptyset\\}}}{\mu([K+x])}\mathrm{d}s\mathrm{d}\theta$
$\displaystyle=\int_{S_{+}^{n-1}}\int_{\mathbb{R}}\frac{\mathds{1}{\\{\operatorname{V}_{n-1}(K\cap
T_{\theta,s-\langle x,\theta\rangle})\leq z\\}}\mathds{1}{\\{K\cap
T_{\theta,s-\langle
x,\theta\rangle}\neq\emptyset\\}}}{\mu([K+x])}\mathrm{d}s\mathrm{d}\theta$
(17)
$\displaystyle=\int_{S_{+}^{n-1}}\int_{\mathbb{R}}\frac{\mathds{1}{\\{\operatorname{V}_{n-1}(K\cap
T_{\theta,t})\leq z\\}}\mathds{1}{\\{K\cap
T_{\theta,t}\neq\emptyset\\}}}{\mu([K+x])}\mathrm{d}t\mathrm{d}\theta$

In (17) we use the translation invariance of the Lebesgue measure. The final
step is obtained by substituting $t=s-\langle x,\theta\rangle$. Via the same
substitution it can be shown that $\mu([K+x])=\mu([K])$. As a result we
obtain: $G_{K+x}(z)=G_{K}(z)$. Moving on to the rotation invariance, let
$M\in\operatorname{SO}(n)$, then the following can be shown:

$MK\cap T_{\theta,s}=M(K\cap T_{M^{T}\theta,s}).$ (18)

Using this, we find:

$\displaystyle G_{MK}(z)$
$\displaystyle=\frac{1}{2}\int_{S^{n-1}}\int_{\mathbb{R}}\frac{\mathds{1}{\\{\operatorname{V}_{n-1}(MK\cap
T_{\theta,s})\leq z\\}}\mathds{1}{\\{MK\cap
T_{\theta,s}\neq\emptyset\\}}}{\mu([MK])}\mathrm{d}s\mathrm{d}\theta$ (19)
$\displaystyle{=}\frac{1}{2}\int_{S^{n-1}}\int_{\mathbb{R}}\frac{\mathds{1}{\\{\operatorname{V}_{n-1}(M(K\cap
T_{M^{T}\theta,s}))\leq z\\}}\mathds{1}{\\{M(K\cap
T_{M^{T}\theta,s})\neq\emptyset\\}}}{\mu([MK])}\mathrm{d}s\mathrm{d}\theta$
$\displaystyle=\frac{1}{2}\int_{S^{n-1}}\int_{\mathbb{R}}\frac{\mathds{1}{\\{\operatorname{V}_{n-1}(K\cap
T_{M^{T}\theta,s})\leq z\\}}\mathds{1}{\\{K\cap
T_{M^{T}\theta,s}\neq\emptyset\\}}}{\mu([MK])}\mathrm{d}s\mathrm{d}\theta$
(20)
$\displaystyle=\int_{S_{+}^{n-1}}\int_{\mathbb{R}}\frac{\mathds{1}{\\{\operatorname{V}_{n-1}(K\cap
T_{u,s})\leq z\\}}\mathds{1}{\\{K\cap
T_{u,s}\neq\emptyset\\}}}{\mu([MK])}\mathrm{d}s\mathrm{d}u.$ (21)

In (19) we use the fact that the inner integral does not change if we replace
$\theta$ with $-\theta$, therefore we may integrate over $S^{n-1}$ instead and
divide the result by two. In (20) we use the rotation invariance of the
Lebesgue measure. In (21) the substitution: $u=M^{T}\theta$ is applied.
Because $M$ is an orthogonal matrix of determinant one, the Jacobian
corresponding to the transformation has determinant one. Since
$M^{T}S^{n-1}=S^{n-1}$, the transformation does not affect the integration
region. Via the same substitutions it can be shown that $\mu([MK])=\mu([K])$
such that indeed $G_{MK}(z)=G_{K}(z)$. Next, we consider scaling of convex
bodies. Let $\lambda>0$, we remark that the following holds:

$\lambda K\cap T_{\theta,s}=\lambda\left(K\cap
T_{\theta,\frac{s}{\lambda}}\right).$ (22)

Using this and the fact that $\operatorname{V}_{n}(\lambda
K)=\lambda^{n}\operatorname{V}_{n}(K)$ for $K\in\mathcal{K}^{n}$, it is once
again a matter of applying a substitution to obtain:

$\displaystyle G_{\lambda
K}(z)=\lambda\int_{S_{+}^{n-1}}\int_{\mathbb{R}}\frac{\mathds{1}{\\{\operatorname{V}_{n-1}(K\cap
T_{\theta,t})\leq\frac{z}{\lambda^{n-1}}\\}}\mathds{1}{\\{K\cap
T_{\theta,t}\neq\emptyset\\}}}{\mu([\lambda K])}\mathrm{d}t\mathrm{d}\theta.$

And similarly, via substitution we find: $\mu([\lambda K])=\lambda\mu([K])$
such that indeed: $G_{\lambda K}(z)=G_{K}(z/\lambda^{n-1})$. The final
statement of the lemma can be seen immediately by recognizing that for all
$\theta\in S_{+}^{n-1}$ and $s\in\mathbb{R}$:

$\operatorname{V}_{n-1}(K\cap T_{\theta,s})\leq\operatorname{V}_{n-1}(L\cap
T_{\theta,s}).$

∎

###### Proof of Lemma 2.

Let $x\in\partial K$. Since $x\in\operatorname{int}L$ there exists a $R>0$
such that $B(x,R)\subset\operatorname{int}L$. Because $x\in\partial K$ we know
that $B(x,R)\cap(\mathbb{R}^{n}\setminus K)\neq\emptyset$. Choose $y\in
B(x,R)\cap(\mathbb{R}^{n}\setminus K)$. Note that $y\in\operatorname{int}L$
and $\operatorname{int}L$ is open. Choose $r_{1}>0$ such that
$B(y,r_{1})\subset\operatorname{int}L$. Because $K$ is closed,
$\mathbb{R}^{n}\setminus K$ is open. Choose $r_{2}>0$ such that
$B(y,r_{2})\subset\mathbb{R}^{n}\setminus K$. Let $r=\min\\{r_{1},r_{2}\\}$,
then $B(y,r)\subset(\operatorname{int}L)\setminus K$ and this ball has a
strictly positive volume. Hence, we find:

$\displaystyle\operatorname{V}_{n}(L)=\operatorname{V}_{n}(L\setminus
K)+\operatorname{V}_{n}(K)\geq\operatorname{V}_{n}(B(y,r))+\operatorname{V}_{n}(K)>\operatorname{V}_{n}(K).$

∎

## Acknowledgements

This research was carried out under project number T17019s in the framework of
the Research Program of the Materials innovation institute (M2i) (www.m2i.nl)
supported by the Dutch government. We thank Kees Bos, Jilt Sietsma and Karo
Sedighiani for fruitful discussions.

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lemma@alttheorem prop@alttheorem claim@alttheorem corollary@alttheorem
definition@alttheorem example@alttheorem remark@alttheorem

# Global dynamics of the $2d$ NLS with white noise potential and generic
polynomial nonlinearity

N. Tzvetkov and N. Visciglia  N. Tzvetkov, CY Cergy-Paris Université, Cergy-
Pontoise, F-95000, UMR 8088 du CNRS<EMAIL_ADDRESS>N. Visciglia,
Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo, 5,
56100 Pisa, Italy<EMAIL_ADDRESS>
###### Abstract.

Using an approach introduced by Hairer-Labbé we construct a unique global
dynamics for the NLS on $\mathbb{T}^{2}$ with a white noise potential and an
arbitrary polynomial nonlinearity. We build the solutions as a limit of
classical solutions (up to a phase shift) of the same equation with smoothed
potentials. This is an improvement on previous contributions of us and
Debussche-Weber dealing with quartic nonlinearities and cubic nonlinearities
respectively.

N.T. is supported by ANR grant ODA (ANR-18-CE40-0020-01), N.V. is supported by
the project PRIN grant 2020XB3EFL and he acknowledge the Gruppo Nazionale per
l’ Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the
Istituzione Nazionale di Alta Matematica (INDAM)

##  1\. Introduction

The aim of this work is to extend the result of [10] to an arbitrary
polynomial nonlinearity. As announced in [10] this will require, in addition
to the modified energies introduced in [10], a suitable use of the dispersive
effect.

We therefore aim to solve, in a sense to be defined, the following Cauchy
problem

(1.1) $i\partial_{t}u=\Delta u+\xi u-u|u|^{p},\quad
u(0,x)=u_{0}(x),\quad(t,x)\in\mathbb{R}\times\mathbb{T}^{2},$

where $p\geq 2$ measures the strength of the nonlinear interaction and
$\xi(x,\omega)$ is the (zero mean value) space white noise which can be seen
as the distribution of the random Fourier series

$\xi(x,\omega)=\sum_{n\in\mathbb{Z}^{2},n\neq 0}\,g_{n}(\omega)\,e^{in\cdot
x}\,,$

where $g_{n}(\omega)$ are standard complex gaussians such that
$\overline{g_{n}}(\omega)=g_{-n}(\omega)$ and otherwise independent.

Thanks to the work by Bourgain [1], we know how to construct the global
dynamics of (1.1) if $\xi$ is replaced by a smooth potential. Therefore a
natural way to solve (1.1) is to regularize $\xi$ and to try to pass to a
limit in the regularized problems. As shown in [3, 10] such a passage to limit
is possible for $p\leq 3$ but only for well-prepared initial data. Therefore,
we are interested in the solutions to the following regularization of (1.1)

(1.2) $i\partial_{t}u_{\varepsilon}=\Delta
u_{\varepsilon}+\xi_{\varepsilon}(x,\omega)u_{\varepsilon}-u_{\varepsilon}|u_{\varepsilon}|^{p}\,\,,\quad
u_{\varepsilon}(0,x)=u_{0}(x)e^{Y(x,\omega)-Y_{\varepsilon}(x,\omega)}\,\,,$

where $\xi_{\varepsilon}=\chi_{\varepsilon}\ast\xi$, $\varepsilon\in(0,1)$ is
a regularization of $\xi$ with
$\chi_{\varepsilon}(x)=\varepsilon^{-2}\chi(x/\varepsilon)$, where $\chi(x)$
is smooth with a support in $\\{|x|<1/2\\}$ and $\int_{\mathbb{T}^{2}}\chi
dx=1$. As in [3, 10], in (1.2), $Y=\Delta^{-1}\xi$ and
$Y_{\varepsilon}=\Delta^{-1}\xi_{\varepsilon}$ is its regularization .

The main result of the paper is the following one, which is an extension of
the one proved in [10] where we were restricted to the powers $p\in[2,3]$.

######  Theorem 1.1.

Assume $p\geq 2$ and $u_{0}(x)$ be such that $e^{Y(x,\omega)}u_{0}(x)\in
H^{2}(\mathbb{T}^{2})$ a.s. Then there exists an event $\Sigma\subset\Omega$
such that $p(\Sigma)=1$ and for every $\omega\in\Sigma$ there exists

$v(t,x,\omega)\in\bigcap_{\gamma\in[0,2)}{\mathcal{C}}(\mathbb{R};H^{\gamma}(\mathbb{T}^{2}))$

such that for every $T>0$ and $\gamma\in[0,2)$ we have:

(1.3)
$\sup_{t\in[-T,T]}\|e^{-iC_{\varepsilon}t}e^{Y_{\varepsilon}(x,\omega)}u_{\varepsilon}(t,x,\omega)-v(t,x,\omega)\|_{H^{\gamma}(\mathbb{T}^{2})}\overset{\varepsilon\rightarrow
0}{\longrightarrow}0,$

where $C_{\varepsilon}=\mathbb{E}(|\nabla Y_{\varepsilon}(x,\omega)|^{2})$
(this quantity is independent of $x$) and $u_{\varepsilon}(t,x,\omega)$ are
solutions to (1.2). Moreover for $\gamma\in[0,1)$ and $\omega\in\Sigma$ we
have

(1.4)
$\sup_{t\in[-T,T]}\big{\|}|u_{\varepsilon}(t,x,\omega)|-e^{-Y(x,\omega)}|v(t,x,\omega)|\big{\|}_{{H^{\gamma}(\mathbb{T}^{2})}\cap
L^{\infty}(\mathbb{T}^{2})}\overset{\varepsilon\rightarrow
0}{\longrightarrow}0.$

The proof of Theorem 1.1 crucially relies on the modified energies and some
results from our previous paper [10]. This makes that the present paper is not
self-contained. As announced in [10], the new ingredient allowing to deal with
general nonlinearities is the use of the dispersive effect which leads to
Proposition 3 below. Proposition 3 displays a gain of regularity with respect
to the Sobolev inequality used in [10], once a time averaging is performed.
Note that we allow logarithmic losses in $\varepsilon$ in our dispersive
bounds. As already used in [3] these losses can be compensated by the
polynomial in $\varepsilon$ convergence of $Y_{\varepsilon}$ to $Y$ in the
natural norms.

In [4, 8] a different approach to the study of (1.1) is introduced. This
approach is based on the construction of a suitable self adjoint realization
of $\Delta+\xi$. Then the initial data in (1.1) is chosen in the domain of
this self adjoint operator. Being in the domain of this self adjoint operator
is the substitute of our assumption of well prepared initial data
$e^{Y(x,\omega)}u_{0}(x)\in H^{2}(\mathbb{T}^{2})$. At the best of our
knowledge the present paper is the first one where global well-posedness is
proved for (1.1) with an arbitrary polynomial nonlinearity $p$, extending the
papers [3] and [10]. Our proof is based on the approach introduced by Hairer-
Labbé in [6].

For the sake of simplicity, we prove Theorem 1.1 in the context of the flat
torus $\mathbb{T}^{2}$. However, it is quite likely that a similar result
holds in the context of a general compact riemannian boundaryless manifolds.
Indeed, the dispersive estimates can be extended to this setting in a
relatively straightforward way. The stochastic analysis results from [10] can
also be extended to this setting by some slightly more involved elaborations.
We will address this question and some related issues in a forthcoming work.

Following Hairer-Labbé [6], we set

(1.5)
$v_{\varepsilon}(t,x,\omega)=e^{-iC_{\varepsilon}t}e^{Y_{\varepsilon}(x,\omega)}u_{\varepsilon}(t,x,\omega)\,,$

where $C_{\varepsilon}$ is the constant appearing in Theorem 1.1. Then
$v_{\varepsilon}$ solves

(1.6) $i\partial_{t}v_{\varepsilon}=\Delta v_{\varepsilon}-2\nabla
v_{\varepsilon}\cdot\nabla Y_{\varepsilon}(x,\omega)+v_{\varepsilon}:|\nabla
Y_{\varepsilon}|^{2}:(x,\omega)-e^{-pY_{\varepsilon}}v_{\varepsilon}|v_{\varepsilon}|^{p},$

where

$:|\nabla Y_{\varepsilon}|^{2}:(x,\omega)=|\nabla
Y_{\varepsilon}|^{2}(x,\omega)-C_{\varepsilon}.$

Following Section 6 in [10] the proof of Theorem 1.1 follows from the next
theorem concerning the behavior of $v_{\varepsilon}(t,x,\omega)$, where
$:|\nabla Y|^{2}:(x,\omega)$ is the renormalized potential defined in [10].

######  Theorem 1.2.

Assume $p\geq 2$ and $u_{0}(x)$ be such that $e^{Y(x,\omega)}u_{0}(x)\in
H^{2}(\mathbb{T}^{2})$ a.s. Then there exists an event $\Sigma\subset\Omega$
such that $p(\Sigma)=1$ and for every $\omega\in\Sigma$ there exists

$v(t,x,\omega)\in\bigcap_{\gamma\in[0,2)}{\mathcal{C}}(\mathbb{R};H^{\gamma}(\mathbb{T}^{2}))$

such that for every fixed $T>0$ and $\gamma\in[0,2)$ we have:

$\sup_{t\in[-T,T]}\|v_{\varepsilon}(t,x,\omega)-v(t,x,\omega)\|_{H^{\gamma}(\mathbb{T}^{2})}\overset{\varepsilon\rightarrow
0}{\longrightarrow}0.$

Here we have denoted by $v_{\varepsilon}(t,x,\omega)$ for $\omega\in\Sigma$
the unique global solution in the space
${\mathcal{C}}(\mathbb{R};H^{2}(\mathbb{T}^{2}))$ of the following problem:

(1.7) $i\partial_{t}v_{\varepsilon}=\Delta v_{\varepsilon}-2\nabla
v_{\varepsilon}\cdot\nabla Y_{\varepsilon}(x,\omega)+v_{\varepsilon}:|\nabla
Y_{\varepsilon}|^{2}:(x,\omega)-e^{-pY_{\varepsilon}}v_{\varepsilon}|v_{\varepsilon}|^{p},\\\
v_{\varepsilon}(0,x)=v_{0}(x)\in H^{2}(\mathbb{T}^{2})$

and $v(t,x,\omega)$ denotes for $\omega\in\Sigma$ the unique global solution
in the space ${\mathcal{C}}(\mathbb{R};H^{\gamma}(\mathbb{T}^{2}))$, for
$\gamma\in(1,2)$, of the following limit problem:

(1.8) $i\partial_{t}v=\Delta v-2\nabla v\cdot\nabla Y(x,\omega)+v:|\nabla
Y|^{2}:(x,\omega)-e^{-pY}v|v|^{p},\,\\\ v(0,x)=v_{0}(x)\in
H^{2}(\mathbb{T}^{2})$

where in both Cauchy problems (1.7) and (1.8)
$v_{0}(x)=e^{Y(x,\omega)}u_{0}(x)$, $\omega\in\Sigma$.

Notations For every $s\in\mathbb{R}$ we denote $s^{+}$ any number belonging to
$(s,s+\delta)$ for a suitable $\delta>0$, similarly $s^{-}$ denotes any number
in $(s-\delta,s)$ for a suitable $\delta>0$. We shall denote by
$L^{p},H^{s},W^{s,p}$ the functional spaces
$L^{p}(\mathbb{T}^{2}),H^{s}(\mathbb{T}^{2}),W^{s,p}(\mathbb{T}^{2})$. In the
sequel we shall denote by $C$ any deterministic finite constant that can
change from line to line and by $C(\omega)$ any random variable defined on
$\Omega$ and finite a.s. We shall denote by $C(\omega,T)$ a constant which is
increasing w.r.t. $T$ and finite for every
$(\omega,T)\in\Sigma\times\mathbb{R}^{+}$ for a suitable event
$\Sigma\subset\Omega$ of full measure. In the rest of the paper for shortness
we will drop writing the $\omega$ dependence of $v_{\varepsilon}$ and
$Y_{\varepsilon}$. For every $a,b$ we denote by $\int_{a}^{b}$ the integral
w.r.t. time variable and $\int_{\mathbb{T}^{2}}$ the integral on
$\mathbb{T}^{2}$.

##  2\. Preliminary facts

We collect in this section some facts proved in [10] and some useful
consequences that will be needed in the sequel.

######  Proposition .

We have the following bound:

(2.1) $\sup_{\varepsilon\in(0,1)}\|Y_{\varepsilon}(x)\|_{L^{\infty}}\leq
C(\omega)\,,$ (2.2) $\sup_{\varepsilon\in(0,1)}\|\nabla
Y_{\varepsilon}(x)\|_{L^{p}}\leq C(\omega)|\log\varepsilon|,\quad\forall
p\in[1,\infty)\,,$ (2.3) $\sup_{\varepsilon\in(0,1)}\|:|\nabla
Y_{\varepsilon}|^{2}:(x)\|_{L^{p}}\leq
C(\omega)|\log\varepsilon|^{2}\,,\quad\forall p\in[1,\infty)\,.$

For every $T>0$ we have the following estimates for the solutions
$v_{\varepsilon}(t,x)$ of (1.7):

(2.4)
$\sup_{\varepsilon\in(0,1)}\|v_{\varepsilon}(t,x)\|_{L^{\infty}((0,T);H^{1})}\leq
C(\omega),$ (2.5)
$\sup_{\varepsilon\in(0,1)}\|v_{\varepsilon}(t,x)\|_{L^{\infty}((0,T);H^{1^{+}})}\leq
C(\omega)\|v_{\varepsilon}\|_{L^{\infty}((0,T);H^{2})}^{0^{+}}\,,$ (2.6)
$\sup_{\varepsilon\in(0,1)}\|v_{\varepsilon}(t,x)\|_{L^{\infty}((0,T);H^{2})}\leq
C(\omega)+C(\omega)\|e^{-Y_{\varepsilon}}\Delta
v_{\varepsilon}\|_{L^{\infty}((0,T);L^{2})}.$

###### Proof.

The bounds (2.1), (2.2), (2.3) have been established in [10] as well as (2.4).
The estimate (2.5) follows by combining interpolation and (2.4), (2.6) follows
by combining elliptic regularity with (2.1). ∎

Next we introduce the family of operators:

(2.7) $H_{\varepsilon}u=\Delta u-2\nabla u\cdot\nabla
Y_{\varepsilon}(x)+u:|\nabla Y_{\varepsilon}|^{2}:(x),$

where as usual we drop the $\omega$ dependence of the operators
$H_{\varepsilon}$. In the sequel we shall need the following result.

######  Proposition .

We have the bound:

(2.8) $\|(H_{\varepsilon}-\Delta)u\|_{L^{2}}\leq
C(\omega)|\ln\varepsilon|^{C}\|u\|_{H^{1^{+}}}.$

###### Proof.

It is sufficient to show the bounds

(2.9) $\|\nabla u\cdot\nabla Y_{\varepsilon}\|_{L^{2}}\leq
C(\omega)\|u\|_{H^{1^{+}}},\quad\|u:|\nabla
Y_{\varepsilon}|^{2}:\|_{L^{2}}\leq C(\omega)\|u\|_{H^{1^{+}}}.$

We have for every $\delta\in(0,1)$

$\|\nabla u\cdot\nabla Y_{\varepsilon}\|_{L^{2}}\leq C\|\nabla
Y_{\varepsilon}\|_{L^{\frac{2}{\delta}}}\|\nabla
u\|_{L^{\frac{2}{1-\delta}}}\leq
C(\omega)|\ln\varepsilon|\|u\|_{H^{1+\delta}}$

where we have used (2.2) and the embedding $H^{\delta}\subset
L^{\frac{2}{1-\delta}}$. The second bound in (2.9) follows by a similar
argument

$\|u:|\nabla Y_{\varepsilon}|^{2}:\|_{L^{2}}\leq C\|:|\nabla
Y_{\varepsilon}|^{2}:\|_{L^{\frac{2}{\delta}}}\|u\|_{L^{\frac{2}{1-\delta}}}\leq
C(\omega)|\ln\varepsilon|^{2}\|u\|_{H^{1+\delta}}$

where we have used (2.3) and $H^{1+\delta}\subset L^{\frac{2}{1-\delta}}$. ∎

##  3\.  A priori bounds of $v_{\varepsilon}$

We introduce the propagator $S_{\varepsilon}(t)$ associated with the linear
problem $i\partial_{t}u=H_{\varepsilon}u,$ where $H_{\varepsilon}$ is defined
in (2.7). The main point of this section is Proposition 3. In order to prove
it, we shall need Strichartz estimates with loss for the propagator
$S_{\varepsilon}(t)$.

######  Proposition .

For every $T>0$ we have the following bound:

(3.1) $\|S_{\varepsilon}(t)\varphi\|_{L^{\infty}((0,T);H^{s})}\leq
C(\omega)|\log\varepsilon|^{C}\|\varphi\|_{H^{s}},\quad s\in[0,2].$

Moreover for every $r,q\in(2,\infty)$ such that $\frac{2}{r}+\frac{2}{q}=1$ we
have

(3.2) $\|S_{\varepsilon}(t)\varphi\|_{L^{r}((0,T);L^{q})}\leq
C(\omega,T)|\log\varepsilon|^{C}\|\varphi\|_{H^{\frac{1}{r}^{+}}}.$

###### Proof.

Estimate (3.1) is established in [3]. For the proof of (3.2), we follow the
argument of [8] which is closely related to the analysis in [2, 7, 9, 11]. The
basic strategy is to perform a perturbative argument with respect to the
evolution $\exp(it\Delta)$ by a partition on small time intervals which makes
the perturbation $H_{\varepsilon}-\Delta$ better but which losses some
regularity on the data because of the summation on the small time intervals.
An additional difficulty resolved in [8] is coming from the fact that a
frequency localisation of $(H_{\varepsilon}-\Delta)(u)$ does not imply a
frequency localisation of $u$.

Let

${\rm Id}=\sum_{N-{\rm dyadic}}\Delta_{N}$

be a Littlewood-Paley partition of the unity. Therefore the issue is to bound

(3.3)
$\|\Delta_{N_{1}}S_{\varepsilon}(t)\Delta_{N_{2}}\varphi\|_{L^{r}((0,T);L^{q})}\,.$

In order to evaluate (3.3), we distinguish two cases according to the sizes of
$N_{1}$ and $N_{2}$ and to sum up on $N_{1},N_{2}$.

First case: $N_{1}\geq N_{2}$

In this case we split the interval $[0,T]$ in an essentially disjoint union of
intervals of size $N_{1}^{-1}$ as

(3.4) $[0,T]=\bigcup_{j}I_{j}$

and we aim to estimate
$\|\Delta_{N_{1}}S_{\varepsilon}(t)\Delta_{N_{2}}\varphi\|_{L^{r}(I_{j};L^{q})}\,.$
Suppose that $I_{j}=[a,b]$. Then following [8] (see also [5]), for $t\in[a,b]$
we can write

(3.5)
$\Delta_{N_{1}}S_{\varepsilon}(t)\Delta_{N_{2}}\varphi=\Delta_{N_{1}}e^{i(t-a)\Delta}S_{\varepsilon}(a)\Delta_{N_{2}}\varphi\\\
+i\int_{a}^{t}\Delta_{N_{1}}e^{i(t-\tau)\Delta}(H_{\varepsilon}-\Delta)S_{\varepsilon}(\tau)\Delta_{N_{2}}\varphi
d\tau.$

We now estimate each term in the right hand-side of (3.5). Using [2], we
estimate the first term as follows for $\delta>0$:

$\|\Delta_{N_{1}}e^{i(t-a)\Delta}S_{\varepsilon}(a)\Delta_{N_{2}}\varphi\|_{L^{r}(I_{j};L^{q})}\leq
CN_{1}^{-\frac{1}{r}-\delta}\|S_{\varepsilon}(a)\Delta_{N_{2}}\varphi\|_{H^{\frac{1}{r}+\delta}}\\\
\leq
C(\omega)|\log\varepsilon|^{C}N_{1}^{-\frac{1}{r}-\delta}\|\varphi\|_{H^{\frac{1}{r}+\delta}}$

where we have used (3.1). Now we estimate the second term in the right hand-
side of (3.5). Using the Minkowski inequality and [2], we can write for every
$\delta>0$:

$\Big{\|}\int_{a}^{t}\Delta_{N_{1}}e^{i(t-\tau)\Delta}(H_{\varepsilon}-\Delta)S_{\varepsilon}(\tau)\Delta_{N_{2}}\varphi
d\tau\Big{\|}_{L^{r}(I_{j};L^{q})}\\\ \leq
C\int_{I_{j}}\|(H_{\varepsilon}-\Delta)S_{\varepsilon}(\tau)\Delta_{N_{2}}\varphi\|_{L^{2}}d\tau\\\
\leq
C(\omega)|\log\varepsilon|^{C}N_{1}^{-1}N_{2}^{1+\frac{\delta}{2}}N_{2}^{-\frac{1}{r}-\delta}\|\varphi\|_{H^{\frac{1}{r}+\delta}}$

where we have used (2.8) and (3.1). Summarizing we get

$\|\Delta_{N_{1}}S_{\varepsilon}(t)\Delta_{N_{2}}\varphi\|_{L^{r}(I_{j};L^{q})}\leq
C(\omega)|\log\varepsilon|^{C}\big{(}N_{1}^{-\frac{1}{r}-\delta}+N_{1}^{-1}N_{2}^{1-\frac{1}{r}-\frac{\delta}{2}}\big{)}\|\varphi\|_{H^{\frac{1}{r}+\delta}}$

and hence using that the number of $I_{j}$ is smaller than $TN_{1}$ taking the
$r$’th power of the previous bound and summing on $j$, we get the estimate

$\|\Delta_{N_{1}}S_{\varepsilon}(t)\Delta_{N_{2}}\varphi\|_{L^{r}((0,T);L^{q})}\leq
C(\omega)T^{\frac{1}{r}}|\log\varepsilon|^{C}\big{(}N_{1}^{-\delta}+N_{1}^{-1+\frac{1}{r}}N_{2}^{1-\frac{1}{r}-\frac{\delta}{2}}\big{)}\|\varphi\|_{H^{\frac{1}{r}+\delta}}$

and hence

(3.6) $\sum_{N_{2}\leq
N_{1}}\|\Delta_{N_{1}}S_{\varepsilon}(t)\Delta_{N_{2}}\varphi\|_{L^{r}((0,T);L^{q})}\leq
C(\omega)T^{\frac{1}{r}}|\log\varepsilon|^{C}\|\varphi\|_{H^{\frac{1}{r}+\delta}}$

where we have used

$\sum_{N_{2}\leq
N_{1}}\big{(}N_{1}^{-\delta}+N_{1}^{-1+\frac{1}{r}}N_{2}^{1-\frac{1}{r}-\frac{\delta}{2}}\big{)}<\infty.$

Second case: $N_{1}\leq N_{2}$

We consider again the splitting (3.4) but this time the intervals $I_{j}$ are
of size $N_{2}^{-1}$. Again we consider (3.5) and we estimate each term of the
right hand-side. Since $N_{2}^{-1}\leq N_{1}^{-1}$, using [2] and (3.1), we
estimate the first term at the right hand-side of (3.5) as

$\|\Delta_{N_{1}}e^{i(t-a)\Delta}S_{\varepsilon}(a)\Delta_{N_{2}}\varphi\|_{L^{r}(I_{j};L^{q})}\leq
C(\omega)|\log\varepsilon|^{C}N_{2}^{-\frac{1}{r}-\delta}\|\varphi\|_{H^{\frac{1}{r}+\delta}}\,,$

where $\delta>0$. Next, as above, we can estimate the second term at the right
hand-side of (3.5) as

$\Big{\|}\int_{a}^{t}\Delta_{N_{1}}e^{i(t-\tau)\Delta}(H_{\varepsilon}-\Delta)S_{\varepsilon}(\tau)\Delta_{N_{2}}\varphi
d\tau\Big{\|}_{L^{r}(I_{j};L^{q})}\\\ \leq
C(\omega)|\log\varepsilon|^{C}N_{2}^{-1}N_{2}^{1+\frac{\delta}{2}}N_{2}^{-\frac{1}{r}-\delta}\|\varphi\|_{H^{\frac{1}{r}+\delta}}\,.$

Summarizing we get

$\|\Delta_{N_{1}}S_{\varepsilon}(t)\Delta_{N_{2}}\varphi\|_{L^{r}(I_{j};L^{q})}\leq
C(\omega)|\log\varepsilon|^{C}\big{(}N_{2}^{-\frac{1}{r}-\delta}+N_{2}^{-\frac{1}{r}-\frac{\delta}{2}}\big{)}\|\varphi\|_{H^{\frac{1}{r}+\delta}}$

and as above, using that the number of $I_{j}$ is smaller than $TN_{2}$ taking
the $r$’th power of the previous bound and summing on $j$, we get the estimate

$\|\Delta_{N_{1}}S_{\varepsilon}(t)\Delta_{N_{2}}\varphi\|_{L^{r}((0,T);L^{q})}\leq
C(\omega)T^{\frac{1}{r}}|\log\varepsilon|^{C}\big{(}N_{2}^{-\delta}+N_{2}^{-\frac{\delta}{2}}\big{)}\|\varphi\|_{H^{\frac{1}{r}+\delta}}.$

Hence we get

(3.7) $\sum_{N_{1}\leq
N_{2}}\|\Delta_{N_{1}}S_{\varepsilon}(t)\Delta_{N_{2}}\varphi\|_{L^{r}((0,T);L^{q})}\leq
C(\omega)T^{\frac{1}{r}}|\log\varepsilon|^{C}\|\varphi\|_{H^{\frac{1}{r}+\delta}}$

since

$\sum_{N_{1}\leq
N_{2}}\big{(}N_{2}^{-\delta}+N_{2}^{-\frac{\delta}{2}}\big{)}<\infty.$

We conclude by combining (3.6) and (3.7) with the Minkowski inequality. ∎

As a consequence we get the following result.

######  Proposition .

For every $T>0$ we have the following estimates:

(3.8) $\|S_{\varepsilon}(t)\varphi\|_{L^{4}((0,T);W^{\frac{3}{4}^{-},4})}\leq
C(\omega,T)|\log\varepsilon|^{C}\|\varphi\|_{H^{1}}$

and

(3.9)
$\Big{\|}\int_{0}^{t}S_{\varepsilon}(t-s)f(s)ds\Big{\|}_{L^{4}((0,T);W^{\frac{3}{4}^{-},4})}\leq
C(\omega,T)|\log\varepsilon|^{C}\|f\|_{L^{1}((0,T);H^{1})}\,.$

###### Proof.

Notice that (3.11) follows by combining (3.10) with the Minkowski inequality.
Next we focus on the proof of (3.10). Notice that for every
$\varepsilon_{0}\in(0,1)$, there exists $q\in(1,\infty)$ such that the
following Gagliardo-Nirenberg inequality occurs:

$\|u\|_{W^{\frac{3}{4}-\varepsilon_{0},4}}\leq
C\|u\|_{L^{q}}^{\frac{1}{2}}\|u\|_{H^{\frac{3}{2}}}^{\frac{1}{2}}$

and hence by integration in time and Hölder inequality in time we get

$\|S_{\varepsilon}(t)\varphi\|_{L^{4}((0,T);W^{\frac{3}{4}-\varepsilon_{0},4})}^{4}\leq
C\|S_{\varepsilon}(t)\varphi\|_{L^{2}((0,T);L^{q})}^{2}\|S_{\varepsilon}(t)\varphi\|_{L^{\infty}((0,T);H^{\frac{3}{2}})}^{2}\\\
\leq
C(\omega)|\log\varepsilon|^{C}\|S_{\varepsilon}(t)\varphi\|_{L^{r}((0,T);L^{q})}^{2}\|\varphi\|_{H^{\frac{3}{2}}}^{2}\leq
C(\omega,T)|\log\varepsilon|^{C}\|\varphi\|_{H^{\frac{1}{2}^{-}}}^{2}\|\varphi\|_{H^{\frac{3}{2}}}^{2}$

where $q,r$ are Strichartz admissible and we have used (3.1), (3.2).

Notice that for initial datum $\varphi=\Delta_{N}\varphi$ which is spectrally
localize at dyadic frequency $N$ we get from the previous bound

$\|S_{\varepsilon}(t)\Delta_{N}\varphi\|_{L^{4}((0,T);W^{\frac{3}{4}-\varepsilon_{0},4})}\leq
C(\omega,T)|\log\varepsilon|^{C}\|\Delta_{N}\varphi\|_{H^{1^{-}}}.$

We conclude (3.10) by summing on $N$. ∎

As a consequence we get the following result.

######  Proposition .

For every $T>0$ we have the following estimates:

(3.10) $\|S_{\varepsilon}(t)\varphi\|_{L^{4}((0,T);W^{\frac{3}{4}^{-},4})}\leq
C(\omega,T)|\log\varepsilon|^{C}\|\varphi\|_{H^{1}}$

and

(3.11)
$\Big{\|}\int_{0}^{t}S_{\varepsilon}(t-s)f(s)ds\Big{\|}_{L^{4}((0,T);W^{\frac{3}{4}^{-},4})}\leq
C(\omega,T)|\log\varepsilon|^{C}\|f\|_{L^{1}((0,T);H^{1})}\,.$

###### Proof.

Notice that (3.11) follows by combining (3.10) with the Minkowski inequality.
Next we focus on the proof of (3.10). Notice that for every
$\varepsilon_{0}\in(0,1)$, there exists $q\in(1,\infty)$ such that the
following Gagliardo-Nirenberg inequality occurs:

$\|u\|_{W^{\frac{3}{4}-\varepsilon_{0},4}}\leq
C\|u\|_{L^{q}}^{\frac{1}{2}}\|u\|_{H^{\frac{3}{2}}}^{\frac{1}{2}}$

and hence by integration in time and Hölder inequality in time we get

$\|S_{\varepsilon}(t)\varphi\|_{L^{4}((0,T);W^{\frac{3}{4}-\varepsilon_{0},4})}^{4}\leq
C\|S_{\varepsilon}(t)\varphi\|_{L^{2}((0,T);L^{q})}^{2}\|S_{\varepsilon}(t)\varphi\|_{L^{\infty}((0,T);H^{\frac{3}{2}})}^{2}\\\
\leq
C(\omega)|\log\varepsilon|^{C}\|S_{\varepsilon}(t)\varphi\|_{L^{r}((0,T);L^{q})}^{2}\|\varphi\|_{H^{\frac{3}{2}}}^{2}\leq
C(\omega,T)|\log\varepsilon|^{C}\|\varphi\|_{H^{\frac{1}{2}^{-}}}^{2}\|\varphi\|_{H^{\frac{3}{2}}}^{2}$

where $q,r$ are Strichartz admissible and we have used (3.1), (3.2).

Notice that for initial datum $\varphi=\Delta_{N}\varphi$ which is spectrally
localize at dyadic frequency $N$ we get from the previous bound

$\|S_{\varepsilon}(t)\Delta_{N}\varphi\|_{L^{4}((0,T);W^{\frac{3}{4}-\varepsilon_{0},4})}\leq
C(\omega,T)|\log\varepsilon|^{C}\|\Delta_{N}\varphi\|_{H^{1^{-}}}.$

We conclude (3.10) by summing on $N$. ∎

Next we get the following bound on the nonlinear solutions $v_{\varepsilon}$
to (1.7).

######  Proposition .

For every $T>0$ we have the following bound:

(3.12) $\|v_{\varepsilon}(t,x)\|_{L^{4}((0,T);W^{\frac{3}{4}^{-},4})}\leq
C(\omega,T)|\log\varepsilon|^{C}(1+\|v_{\varepsilon}(t,x)\|_{L^{\infty}((0,T);H^{2})}^{0^{+}}).$

###### Proof.

By combining Proposition 3 with the integral formulation associated with (1.7)
we get:

$\|v_{\varepsilon}\|_{L^{4}((0,T);W^{\frac{3}{4}^{-},4})}\leq
C(\omega,T)|\log\varepsilon|^{C}\|v_{\varepsilon}(0)\|_{H^{1}}+C(\omega,T)|\log\varepsilon|^{C}\int_{0}^{T}\|e^{-pY_{\varepsilon}}v_{\varepsilon}|v_{\varepsilon}|^{p}\|_{H^{1}}\\\
\leq
C(\omega,T)|\log\varepsilon|^{C}\|v_{\varepsilon}(0)\|_{H^{1}}+C(\omega,T)|\log\varepsilon|^{C}\int_{0}^{T}\|v_{\varepsilon}\|_{H^{1}}\|v_{\varepsilon}\|_{L^{\infty}}^{p}\|e^{-pY_{\varepsilon}}\|_{L^{\infty}}\\\
+C(\omega,T)|\log\varepsilon|^{C}\int_{0}^{T}\|\nabla
Y_{\varepsilon}\|_{L^{2}}\|e^{-pY_{\varepsilon}}\|_{L^{\infty}}\|v_{\varepsilon}\|_{L^{\infty}}^{p+1}$

and we conclude by using the Sobolev embedding $H^{1^{+}}\subset L^{\infty}$,
(2.1), (2.2), (2.4), (2.5). ∎

We conclude this section with the following key estimate.

######  Proposition .

We have the following bound for a suitable $\eta\in(0,1)$ and for every $T>0$:

$\|v_{\varepsilon}(t,x)\|_{L^{2}((0,T);W^{1,4})}^{2}\leq
C(\omega,T)|\log\varepsilon|^{C}(1+\|v_{\varepsilon}(t,x)\|_{L^{\infty}((0,T);H^{2})}^{\eta}).$

###### Proof.

We have the bound for time independent functions:

$\|u\|_{W^{1,4}}\leq
C\|u\|_{W^{\frac{3}{4}^{-},4}}^{\frac{2}{3}^{-}}\|u\|_{H^{2}}^{\frac{1}{3}^{+}}.$

Hence by integration in time and by choosing $u=v_{\varepsilon}$ we get

$\|v_{\varepsilon}\|_{L^{2}((0,T);W^{1,4})}^{2}\leq
CT\|v_{\varepsilon}\|_{L^{\frac{4}{3}^{-}}((0,T);W^{\frac{3}{4}^{-},4})}^{\frac{4}{3}^{-}}\|v_{\varepsilon}\|_{L^{\infty}((0,T);H^{2})}^{\frac{2}{3}^{+}}$
$\leq
CT\|v_{\varepsilon}\|_{L^{4}((0,T);W^{\frac{3}{4}^{-},4})}^{\frac{4}{3}^{-}}\|v_{\varepsilon}\|_{L^{\infty}((0,T);H^{2})}^{\frac{2}{3}^{+}}.$

We conclude by Proposition 3. ∎

##  4\. Proof of Theorem 1.2

We aim at proving the following bound for every given $T>0$:

(4.1)
$\|v_{\varepsilon}(t,x)\|_{L^{\infty}((0,T);H^{2})}\leq|\log\varepsilon|^{C(\omega,T)},\,\,\forall\varepsilon\in(0,\frac{1}{2}).$

Recall that the bound (4.1) has been achieved in [10] in the case $2\leq p\leq
3$ (see Proposition 4.5 in [10]). The main point is that we get the bound
(4.1) for every $p\geq 2$. Once (4.1) is achieved then Theorem 1.2 can be
proved exactly as in [10, Section 5].

We can now establish (4.1). In order to do that we recall some notations from
[10]. Denote by ${\mathcal{H}}_{\varepsilon},{\mathcal{F}}_{\varepsilon}$ and
${\mathcal{G}}_{\varepsilon}$ the energies introduced along [10, Proposition
4.1] which satisfy

(4.2)
$\frac{d}{dt}({\mathcal{F}}_{\varepsilon}(v_{\varepsilon})-{\mathcal{G}}_{\varepsilon}(v_{\varepsilon}))=-{\mathcal{H}}_{\varepsilon}(v_{\varepsilon}).$

An important point is to obtain the following modification of [10, Proposition
4.3] which gains on the power of $\|e^{-Y_{\varepsilon}}\Delta
v_{\varepsilon}\|_{L^{\infty}((0,T);L^{2})}$ appearing in the right hand-side
by exploiting the averaging in the time variable.

######  Proposition .

For a suitable $\gamma\in(1,2)$ we have the bound:

$\int_{0}^{T}|{\mathcal{H}}_{\varepsilon}(v_{\varepsilon}(s))|ds\leq
C(\omega,T)|\log\varepsilon|^{C}+\|e^{-Y_{\varepsilon}}\Delta
v_{\varepsilon}\|_{L^{\infty}((0,T);L^{2})}^{\gamma}\,.$

###### Proof.

By using the Hölder inequality, the Leibnitz rule and the diamagnetic
inequality $|\partial_{t}|u||\leq|\partial_{t}u|$ we get that the first three
terms in ${\mathcal{H}}_{\varepsilon}(v_{\varepsilon})$ can be estimated by:

$\int_{\mathbb{T}^{2}}|\partial_{t}v_{\varepsilon}||\nabla
v_{\varepsilon}|^{2}|v_{\varepsilon}|^{p-1}e^{-(p+2)Y_{\varepsilon}}\leq
C(\omega)\|\partial_{t}v_{\varepsilon}\|_{L^{2}}\|\nabla
v_{\varepsilon}\|_{L^{4}}^{2}\|v_{\varepsilon}\|_{L^{\infty}}^{p-1}.$

where we have used (2.1). By using the equation solved by
$v_{\varepsilon}(t,x)$ and the Sobolev embedding $H^{1^{+}}\subset L^{\infty}$
we get from the estimate above after integration in time:

$\int_{0}^{T}\int_{\mathbb{T}^{2}}|\partial_{t}v_{\varepsilon}||\nabla
v_{\varepsilon}|^{2}|v_{\varepsilon}|^{p-1}e^{-(p+2)Y_{\varepsilon}}\\\ \leq
C(\omega)\|\Delta v_{\varepsilon}\|_{L^{\infty}((0,T);L^{2})}\|\nabla
v_{\varepsilon}\|_{L^{2}((0,T);L^{4})}^{2}\|v_{\varepsilon}\|_{L^{\infty}((0,T);H^{1^{+}})}^{p-1}\\\
+C(\omega)\|\nabla v_{\varepsilon}\cdot\nabla
Y_{\varepsilon}\|_{L^{\infty}((0,T);L^{2})}\|\nabla
v_{\varepsilon}\|_{L^{2}((0,T);L^{4})}^{2}\|v_{\varepsilon}\|_{L^{\infty}((0,T);H^{1^{+}})}^{p-1}\\\
+C(\omega)\|v_{\varepsilon}:|\nabla
Y_{\varepsilon}|^{2}:\|_{L^{\infty}((0,T);L^{2})}\|\nabla
v_{\varepsilon}\|_{L^{2}((0,T);L^{4})}^{2}\|v_{\varepsilon}\|_{L^{\infty}((0,T);H^{1^{+}})}^{p-1}\\\
+C(\omega)\|e^{-pY_{\varepsilon}}v_{\varepsilon}|v_{\varepsilon}|^{p}\|_{L^{\infty}((0,T);L^{2})}\|\nabla
v_{\varepsilon}\|_{L^{2}((0,T);L^{4})}^{2}\|v_{\varepsilon}\|_{L^{\infty}((0,T);H^{1^{+}})}^{p-1}\\\
=I+II+III+IV.$

Combining (2.1), (2.5), (2.6) and Proposition 3 we get

$I\leq C(\omega,T)|\log\varepsilon|^{C}\|\Delta
v_{\varepsilon}\|_{L^{\infty}((0,T);L^{2})}\|v_{\varepsilon}\|_{L^{\infty}((0,T);H^{2})}^{0^{+}}\big{(}1+\|v_{\varepsilon}\|_{L^{\infty}((0,T);H^{2})}^{\eta}\big{)}\\\
\leq C(\omega,T)|\log\varepsilon|^{C}+\|e^{-Y_{\varepsilon}}\Delta
v_{\varepsilon}\|_{L^{\infty}((0,T);L^{2})}^{1+\eta^{+}}.$

By combining now Hölder inequality, (2.5) and Proposition 3 we get

$II\leq C(\omega,T)|\log\varepsilon|^{C}\|\nabla
Y_{\varepsilon}\|_{L^{4}}\|\nabla
v_{\varepsilon}\|_{L^{\infty}((0,T);L^{4})}\big{(}1+\|v_{\varepsilon}\|_{L^{\infty}((0,T);H^{2})}\big{)}^{\eta^{+}}$

and hence by (2.2) and Sobolev embedding $H^{1}\subset L^{4}$ we conclude

$II\leq
C(\omega,T)|\log\varepsilon|^{C}(1+\|v_{\varepsilon}\|_{L^{\infty}((0,T);H^{2})})^{1+\eta^{+}}\\\
\leq C(\omega,T)|\log\varepsilon|^{C}+\|e^{-Y_{\varepsilon}}\Delta
v_{\varepsilon}\|_{L^{\infty}((0,T);L^{2})}^{1+\eta^{+}}$

where we used at the last step (2.6). We also get

$III\leq C(\omega,T)|\log\varepsilon|^{C}+\|e^{-Y_{\varepsilon}}\Delta
v_{\varepsilon}\|_{L^{\infty}((0,T);L^{2})}^{1+\eta^{+}}$

whose proof is identical to the estimate of the term $II$ given above, except
that we use (2.3) instead of (2.2). For the term $IV$ we get by (2.1), (2.4),
(2.5), (2.6) and Proposition 3

$IV\leq
C(\omega,T)|\log\varepsilon|^{C}\|v_{\varepsilon}\|_{L^{\infty}((0,T);L^{2(p+1)})}^{p+1}\big{(}1+\|v_{\varepsilon}\|_{L^{\infty}((0,T);H^{2})})^{\eta^{+}}\\\
\leq C(\omega,T)|\log\varepsilon|^{C}+\|e^{-Y_{\varepsilon}}\Delta
v_{\varepsilon}\|_{L^{\infty}((0,T);L^{2})}^{\eta^{+}},$

where we used at the last step the Sobolev embedding $H^{1}\subset
L^{2(p+1)}$. Concerning the last term in the expression of
${\mathcal{H}}_{\varepsilon}(v_{\varepsilon})$ we can estimate it as follows:

$\int_{\mathbb{T}^{2}}|\partial_{t}v_{\varepsilon}||v_{\varepsilon}|^{p}|\nabla
Y_{\varepsilon}||\nabla v_{\varepsilon}|e^{-(p+2)Y_{\varepsilon}}\\\ \leq
C(\omega)\|v_{\varepsilon}\|_{L^{8p}}^{p}\|\partial_{t}v_{\varepsilon}\|_{L^{2}}\|\nabla
Y_{\varepsilon}\|_{L^{8}}\|\nabla v_{\varepsilon}\|_{L^{4}}\\\ \leq
C(\omega)|\log\varepsilon|\|\partial_{t}v_{\varepsilon}\|_{L^{2}}\|\nabla
v_{\varepsilon}\|_{L^{4}}$

where we have used (2.1), (2.2), the Sobolev embedding $H^{1}\subset L^{8p}$
and (2.4). Next we replace $\partial_{t}v_{\varepsilon}$ by using the equation
solved by $v_{\varepsilon}$ and, thanks to the following time-independent
Gagliardo-Nirenberg inequality

(4.3) $\|\nabla u\|_{L^{4}}^{2}\leq C\|\nabla u\|_{L^{2}}\|\Delta
u\|_{L^{2}},$

we can continue the estimate above as follows:

$\dots\leq C(\omega)|\log\varepsilon|\|\Delta
v_{\varepsilon}\|_{L^{2}}\|\Delta
v_{\varepsilon}\|_{L^{2}}^{\frac{1}{2}}\|\nabla
v_{\varepsilon}\|_{L^{2}}^{\frac{1}{2}}\\\ +C(\omega)|\log\varepsilon|\|\nabla
v_{\varepsilon}\cdot\nabla Y_{\varepsilon}\|_{L^{2}}\|\Delta
v_{\varepsilon}\|_{L^{2}}^{\frac{1}{2}}\|\nabla
v_{\varepsilon}\|_{L^{2}}^{\frac{1}{2}}\\\
+C(\omega)|\log\varepsilon|\|v_{\varepsilon}:|\nabla
Y_{\varepsilon}|^{2}:\|_{L^{2}}\|\Delta
v_{\varepsilon}\|_{L^{2}}^{\frac{1}{2}}\|\nabla
v_{\varepsilon}\|_{L^{2}}^{\frac{1}{2}}\\\
+C(\omega)|\log\varepsilon|\|e^{-pY_{\varepsilon}}v_{\varepsilon}|v_{\varepsilon}|^{p}\|_{L^{2}}\|\Delta
v_{\varepsilon}\|_{L^{2}}^{\frac{1}{2}}\|\nabla
v_{\varepsilon}\|_{L^{2}}^{\frac{1}{2}}$

and by the Sobolev embedding $H^{1}\subset L^{4}$ and (2.1), (2.4)

$\dots\leq C(\omega)|\log\varepsilon|\|\Delta
v_{\varepsilon}\|_{L^{2}}^{\frac{3}{2}}+C(\omega)|\log\varepsilon|\|\nabla
v_{\varepsilon}\|_{L^{4}}\|\nabla Y_{\varepsilon}\|_{L^{4}}\|\Delta
v_{\varepsilon}\|_{L^{2}}^{\frac{1}{2}}\\\
+C(\omega)|\log\varepsilon|\|v_{\varepsilon}\|_{L^{4}}\|:|\nabla
Y_{\varepsilon}|^{2}:\|_{L^{4}}\|\Delta
v_{\varepsilon}\|_{L^{2}}^{\frac{1}{2}}+C(\omega)|\log\varepsilon|\|v_{\varepsilon}|v_{\varepsilon}|^{p}\|_{L^{2}}\|\Delta
v_{\varepsilon}\|_{L^{2}}^{\frac{1}{2}}\\\ \leq
C(\omega)|\log\varepsilon|\|\Delta
v_{\varepsilon}\|_{L^{2}}^{\frac{3}{2}}+C(\omega)|\log\varepsilon|^{2}\|\Delta
v_{\varepsilon}\|_{L^{2}}^{\frac{3}{2}}+C(\omega)|\log\varepsilon|^{3}\|\Delta
v_{\varepsilon}\|_{L^{2}}^{\frac{1}{2}}+C(\omega)|\log\varepsilon|\|\Delta
v_{\varepsilon}\|_{L^{2}}^{\frac{1}{2}}$

where we have used (2.2) and (2.3). Summarizing we get from the computation
above and by (2.6)

$\int_{0}^{T}\int_{\mathbb{T}^{2}}|\partial_{t}v_{\varepsilon}||v_{\varepsilon}|^{p}|\nabla
Y_{\varepsilon}||\nabla v_{\varepsilon}|e^{-(p+2)Y_{\varepsilon}}\leq
C(\omega,T)|\log\varepsilon|^{C}+\|e^{-Y_{\varepsilon}}\Delta
v_{\varepsilon}\|_{L^{2}}^{\frac{3}{2}^{+}}.$

∎

Next we shall also need the following bound from [10, Proposition 4.4].

######  Proposition .

For every $\mu>0$ there exists a random variable $C(\omega)$ such that:

(4.4)
$\big{|}{\mathcal{F}}_{\varepsilon}(v_{\varepsilon})-\int_{\mathbb{T}^{2}}|\Delta
v_{\varepsilon}|^{2}e^{-2Y_{\varepsilon}}\big{|}<\mu\|e^{-Y_{\varepsilon}}\Delta
v_{\varepsilon}\|_{L^{2}}^{2}+C(\omega)|\log\varepsilon|^{4}$

and

(4.5)
$|{\mathcal{G}}_{\varepsilon}(v_{\varepsilon})|<\mu\|e^{-Y_{\varepsilon}}\Delta
v_{\varepsilon}\|_{L^{2}}^{2}+C(\omega)|\log\varepsilon|^{4}.$

We have now all tools to prove (4.1). By integration in time of (4.2) and by
combining Proposition 4 (where we choose $\mu$ small enough in order to absorb
on the l.h.s. the term $\|e^{-Y_{\varepsilon}}\Delta
v_{\varepsilon}\|_{L^{\infty}((0,T);L^{2})}^{2})$ with Proposition 4 we get

$\|e^{-Y_{\varepsilon}}\Delta
v_{\varepsilon}\|_{L^{\infty}((0,T);L^{2})}^{2}\leq
C(\omega,T)|\log\varepsilon|^{C}+\|e^{-Y_{\varepsilon}}\Delta
v_{\varepsilon}\|_{L^{\infty}((0,T);L^{2})}^{\gamma},\quad\gamma<2$

and hence we conclude (4.1).

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