In this section we review the spline-based image segmentation procedure that we have developed in the context of scan-based immersed isogeometric analysis <cit.>. In Section <ref> we first discuss the spline-based level set construction to smoothen scan data. In Section <ref> we review the algorithms used to construct an explicit parametrization of the scan domain. An example is finally shown in Section <ref>, illustrating the effectivity of the topology-preservation procedure developed in Ref. <cit.>.

A two-dimensional illustration of the spline-based segmentation procedure. (a) Grayscale data on a $32 \times 32$ voxel mesh, $\voxelmesh$, as in (<ref>). (b) The voxel segmentation obtained by thresholding the grayscale data as in (<ref>). (c) The smooth level set function obtained using a THB-spline basis constructed on the voxel mesh with local refinements as in (<ref>). The exact immersed boundary, which is in principle unknown, is shown in red for reference.

§.§ B-spline smoothing of the scan data

The spline-based level set construction is illustrated in Fig. <ref>. We consider a $\ndims$-dimensional scan domain, $\imagedomain= [0,L_{1}] \times\ldots \times [0,L_{\ndims}]$ with volume $V_{\rm scan}= \prod_{i=1}^{n_d} L_i$, which is partitioned by a set of $\maxvoxel$ voxels, as illustrated in Fig. <ref>a. We denote the voxel mesh by $\voxelmesh$, with $\boldsymbol{\voxelsize}$ the voxel size in each direction. The grayscale intensity function is then defined as

\begin{equation}
g: \voxelmesh \rightarrow \mathscr{G},
\label{eq:grayscale}
\end{equation}

with $\mathscr{G}$ the range of the grayscale data (e.g., from 0 to 255 for 8 bit unsigned integers). An approximation of the object $\domain$ can be obtained by thresholding the grayscale data,

\begin{equation}
\domain\approx \{ \boldsymbol{x} \in \imagedomain | g(\boldsymbol{x}) > g_{\rm crit} \}\subset \imagedomain,
\label{eq:voxeldomain}
\end{equation}

where $g_{\rm crit}$ is the threshold value. As a consequence of the piecewise definition of the grayscale data in equation (<ref>), the boundary of the segmented object is non-smooth when the grayscale data is segmented directly. In the context of analysis, the non-smoothness of the boundary can be problematic, as irregularities in the surface may lead to non-physical features in the solution to the problem.

The spline-based segmentation procedure developed in Refs. <cit.> enables the construction of a smooth boundary approximation based on voxel data. The key idea of this spline-based segmentation technique is to smoothen the grayscale function (<ref>) by convoluting it using an $n$-dimensional spline basis, $\{ \basisfunc_{i,\basisorder}(\scanmesh) \}_{i=1}^n$, defined over a mesh, $\scanmesh$, with element size, $\boldsymbol{h}=(h_1,\ldots,h_d)$ (note that the mesh size can differ from the voxel size). The order $\basisorder$ of the spline basis functions is assumed to be constant and isotropic. We consider THB-splines <cit.> for the construction of locally refined spaces. By considering full-regularity ($C^{\basisorder-1}$-continuous) splines of degree $\basisorder>1$, a smooth level set approximation of (<ref>) is obtained by the convolution operation

\begin{align}
f(\boldsymbol{x}) &= \sum \limits_{i=1}^{n} \basisfunc_{i,\basisorder} (\boldsymbol{x}) a_{i}, &  a_{i} &= \cfrac{\int_{\imagedomain} \basisfunc_{i,\basisorder} (\boldsymbol{x}) g(\boldsymbol{x}) {\rm d}\boldsymbol{x}}{\int_{\imagedomain} \basisfunc_{i,\basisorder} (\boldsymbol{x}) {\rm d}\boldsymbol{x}}  , \label{eq:bsplinefunc}
\end{align}

where $\{ a_i \}_{i=1}^n$ are the coefficients of the discrete level set function. The smoothed domain then follows by thresholding of this level set function:

\begin{equation}
\domain\approx \{ \boldsymbol{x} \in \imagedomain | f(\boldsymbol{x}) > f_{\rm crit} \}\subset \imagedomain.
\label{eq:implicitdomain}
\end{equation}

The spline level set function corresponding to the voxel data in Fig. <ref>a is illustrated in Fig. <ref>c for the case of a locally refined mesh $\scanmesh$ and second order ($\basisorder=2$) THB-splines. As can be seen, the object retrieved from the convoluted level set function more closely resembles the original geometry in Fig. <ref>a compared to the voxel segmentation in Fig. <ref>b. Also, as a consequence of the higher-order continuity of the spline basis, the boundaries of the domain are smooth, which is in closer agreement with reality.

The convolution operation (<ref>) is computationally efficient, resulting from the fact that it is not required to solve a linear system of equations (in contrast to a (global) $L^2$-projection) and the restricted support of the convolution kernel. Moreover, the convolution strategy has various properties that are advantageous in the context of scan-based immersed isogeometric analysis (see Refs. <cit.> for details):

§.§.§ Conservation of the gray scale intensity

Under the condition that the spline basis, $\{ \basisfunc_{i,\basisorder}(\scanmesh) \}_{i=1}^n$, satisfies the partition of unity property (e.g., B-splines, THB-splines), the smooth level set approximation (<ref>) conserves the gray scale intensity of the original data in the sense that

\begin{align}
\frac{1}{V_{\rm scan}} \int \limits_{\Omega_{\rm scan}} f\,{\rm d}V =  \frac{1}{V_{\rm scan}} \int \limits_{\Omega_{\rm scan}} g\,{\rm d}V = \frac{1}{m_{\rm vox}} \sum_{\element \in \voxelmesh} g(\element).
\label{eq:intensity}
\end{align}

This property ensures that there is a direct relation between the threshold value, $f_{\rm crit}$, for the smooth level set reconstruction (<ref>) and that of the original data, $g_{\rm crit}$, in equation (<ref>).

§.§.§ Local boundedness by the original data

On every voxel $\element \in \voxelmesh$, the level set function (<ref>) is bound by the extrema of the voxel function over the support extension <cit.>, $\widetilde{\element}$, i.e.,

\begin{align}
&\min_{\boldsymbol{x} \in  \widetilde{\element}} g(\boldsymbol{x})  \leq f(\boldsymbol{x}) \leq \max_{\boldsymbol{x} \in \widetilde{\element}} g(\boldsymbol{x}) & &\forall \boldsymbol{x} \in \element.
\label{eq:bounds}
\end{align}

These bounds preclude overshoots and undershoots, which indicates that no spurious oscillations are created by the smoothing procedure (contrasting the case of an $L^2$-projection).

§.§.§ Approximate Gaussian blurring

The spline-based convolution operation (<ref>) can be written as an integral transform

\begin{align}
f(\boldsymbol{x}) &= \int_{\imagedomain}\mathcal{K}(\boldsymbol{x},\boldsymbol{y}) g(\boldsymbol{y}) \:{\rm d} \boldsymbol{y},  &  \mathcal{K}(\boldsymbol{x}, \boldsymbol{y}) &= \sum \limits_{i=1}^{n} \cfrac{\basisfunc_{i,\basisorder}(\boldsymbol{x}) \basisfunc_{i,\basisorder}(\boldsymbol{y})}{\int_{\imagedomain} \basisfunc_{i,\basisorder}(\boldsymbol{z})\:{\rm d}\boldsymbol{z} }, \label{eq:convolution}
\end{align}

where $\mathcal{K}(\boldsymbol{x},\boldsymbol{y})$ is the kernel of the transformation.

The integral transform (<ref>) acts as an approximate Gaussian filter <cit.>. We illustrate this behavior for the case of one-dimensional voxel data, which is smoothed using a B-spline basis defined on a uniform mesh, $\scanmesh$, with mesh size $\basismeshsize$. Following the derivation in Ref. <cit.> – in which the essential step is to approximate the B-spline basis functions by rescaled Gaussians <cit.> – the integration kernel (<ref>) can be approximated by

\begin{equation}
\mathcal{K}(x,y) \approx \kappa( x-y ) = \frac{1}{\sigma \sqrt{2\pi}} \exp{\left( {-\cfrac{(x-y)^2}{2 \sigma^2}} \right)}, \label{eq:kernel}
\end{equation}

where the width of the smoothing kernel is given by $\sigma = \basismeshsize \sqrt{ \frac{\basisorder+1}{6} }$. Next, we consider an object of size $\ell$, represented by the grayscale function

\begin{align}
g(x) &= \begin{cases}
1 & | x |  < \ell/2\\
0 & \mbox{otherwise}
\end{cases}.
\end{align}

This object and the corresponding approximate level set function (<ref>) are illustrated in Fig. <ref> for various feature-size-to-mesh ratios, $\hat{\ell}=\ell/\basismeshsize=2,1,\frac{1}{2}$, and B-spline degrees, $\basisorder=2,3,4$. Following the (Fourier) analysis in Ref. <cit.>, the value of the smoothed level set function at $x=0$ follows as

\begin{equation}
\hat{f}_1(0) = \hat{\ell} \sqrt{\frac{3}{\pi(\basisorder+1)} }  ~ \exp{\left( - \frac{3 \hat{\ell}^2}{16(\basisorder+1)}   \right)},
\label{eq:approxmax}
\end{equation}

which conveys that the maximum value of the smoothed level set depends linearly on the relative feature size $\hat{\ell}$ (for sufficiently small $\hat{\ell}$), and decreases with increasing B-spline order.

Fig. <ref> shows the case for which the considered feature is twice as large as the mesh size, i.e., $\hat{\ell}=2$, illustrating that the sharp boundaries of the original grayscale function are significantly smoothed. The decrease in the maximum level set value as given by equation (<ref>) is observed. When the level set function is segmented by a threshold of $g_{\rm crit}=0.5$, a geometric feature that closely resembles the original one is recovered. Fig. <ref>-fig:spaceloverhhalf illustrate cases where the feature length, $\ell$, is not significantly larger than the mesh size, $\basismeshsize$. For the case where the feature size is equal to the size of the mesh, the maximum of the level set function drops significantly compared to the case of $\hat{\ell}=2$. When considering second-order B-splines, the maximum is still marginally above $g_{\rm crit}=0.5$. Although the recovered feature is considerably smaller than the original one, it is still detected in the segmentation procedure. When increasing the B-spline order, the maximum value of the level set drops below the segmentation threshold, however, indicating that the feature will no longer be detected. When decreasing the feature length further, as illustrated in Fig. <ref>, the feature would be lost when segmentation is performed with $g_{\rm crit}=0.5$, regardless of the order of the spline basis.

The implications of this smoothing behavior of the convolution operation (<ref>) in the context of the spline-based segmentation will be discussed in Section <ref>.

Smoothed level set approximation (<ref>) of a geometric feature in the spatial domain, with $\hat{x}=x/h$, for various feature-size-to-mesh ratios, $\hat{\ell}=\ell/h$, and B-spline degrees.

§.§ Octree-based tessellation procedure

Our scan-based isogeometric analysis approach requires the construction of an explicit parametrization of the implicit level set domain (<ref>). In this section we outline the segmentation procedure that we use to obtain an explicit parametrization of the domain and its (immersed) boundaries. This procedure, which is based on the octree subdivision approach introduced in the context of the Finite Cell Method in Ref. <cit.>, is illustrated in Fig. <ref>. In Section <ref> we first discuss the employed octree procedure, after which the tessellation procedure used at the lowest level of subdivision is detailed in Section <ref>. Without loss of generality, in the remainder we will assume that the level set function $f$ is shifted such that $f_{\rm crit}=0$.

Illustration of the octree-based tessellation procedure to acquire an explicit parametrization of the immersed computational domain. The number of element subdivisions is indicated by $\varrho$, which is equal to the octree depth at the lowest level of bisectioning.

§.§.§ Octree subdivision

In our analysis framework we consider a regular mesh $\scanmesh$ that conforms to the scan domain $\imagedomain$. Each element in this mesh is a hyperrectangle (i.e., a line in one dimension, a rectangle in two dimensions, and a hexahedron in three dimensions) with size ${\boldsymbol{h}}=(h_1,\ldots, h_{\ndims})$ in each direction. Elements, $K$, that are intersected by the immersed boundary are trimmed, resulting in a partitioning, $\mathcal{P}_K$, of a cut element (see Fig. <ref>).

We employ the octree-based trimming procedure outlined in Alg. <ref>. This procedure follows a bottom-up approach <cit.> in which the level set function (<ref>) is sampled at the $2^{\textit{depth}}+1$ vertices of the octree in each direction for each element, where $\textit{depth}$ is the number of subdivision operations performed to detect the immersed boundary.

The trimming procedure takes the evaluated level set values, the subdivision depth and the dimension of the element as input arguments. If all level set values are positive (L<ref>), the trim_element function retains the complete element in the mesh. If all level set values are non-positive (L<ref>), the element is discarded. When some of the level set values are positive and some are non-positive (L<ref>), this implies that the element is intersected by the immersed boundary. In this case, the element is subdivided (bisected) into $2^\ndims$ children (L<ref>). For each child a recursive call to the trim_element function is made (L<ref>).

The recursive subdivision routine is terminated at the lowest level of subdivision, i.e., at $\textit{depth}=0$ (L<ref>). Since our analysis approach requires the evaluation of functions on the immersed boundary, it is convenient to also obtain an explicit parametrization of this boundary. To obtain this parametrization, at the lowest level of subdivision we consider a tessellation procedure based on the level set values. The function get_mosaic_element that implements this procedure is discussed in Section <ref>.

§.§.§ Midpoint tessellation

At the lowest level of subdivision of the octree procedure, we perform a tessellation based on the $2^\ndims$ level set values at that level. In the scope of our work, an important requirement for the tessellation procedure is that it is suitable for the consideration of interface problems. Practically, this means that if the procedure is applied to the negated level set function, a partitioning of the complementary part of the cell is obtained, with an immersed boundary that matches with that of the tessellation based on the original level set function (illustrated in Fig. <ref>). Standard tessellation procedures, specifically Delaunay tessellation <cit.>, do not meet this requirement, as the resulting tessellation is always convex. When an immersed boundary tessellation is convex from one side, it is concave from the other side, meaning that it cannot be identically represented by the Delaunay tessellation. Another complication of Delaunay tessellation is its lack of uniqueness <cit.>, which in our applications results in non-matching interface tessellations.

To overcome the deficiencies associated with Delaunay tessellation, we have developed a dedicated tessellation procedure that suits the needs of our immersed analysis approach <cit.>. We refer to the developed procedure as midpoint tessellation, which is illustrated in Figs. <ref> and <ref> for the two-dimensional and three-dimensional case, respectively. The get_mosaic_element function outlined in Alg. <ref> implements our midpoint tessellation procedure. This function is called by the octree algorithm at the lowest level of subdivision (Alg. <ref>, L<ref>), taking the $2^\ndims$ level set values at the octree vertices as input. The function returns a tessellation of the octree cell that (approximately) fits to the immersed boundary.

To explain Alg. <ref>, we first consider the two-dimensional case, which is illustrated in Fig. <ref>. The midpoint tessellation procedure commences with looping over all the edges of the element and recursively calling the get_mosaic_element function to truncate the edges that are intersected by the immersed boundary (L<ref>-L<ref>, Fig. <ref>). A set of zero_points is then computed by linear interpolation of the level set function across the diagonals between the centroid of the rectangle and its vertices (L<ref>, Fig. <ref>), and the arithmetic average of these points is defined as the midpoint (L<ref>, Fig. <ref>). The tessellation is then created by extruding the (truncated) edges toward this midpoint (Figs. <ref>-<ref>). Note that if this procedure is applied to the negated level set values, a tessellation of the complementary part of the rectangular element with a coincident immersed boundary is obtained (Fig. <ref>).

Since the tessellation algorithm recursively traverses the dimensions of the element, it can directly be extended to the three-dimensional case, as illustrated in Fig. <ref>. In three dimensions, all six faces of the element are tessellated by calling the get_mosaic_element function (L<ref>-L<ref>, Fig. <ref>). Based on the diagonals between the centroid of the element and its vertices, zero level set points (L<ref>) and a corresponding midpoint (L<ref>) are then computed. The three-dimensional tessellation is finally constructed by extrusion of all (truncated) faces toward the midpoint (Fig. <ref>-<ref>). As in the two dimensional case, a conforming interface is obtained when the procedure is applied to the negated level set values.

The algorithms presented in this section are presented for the case of hyperrectangles, i.e., a rectangle in two dimensions and a hexagon in three dimensions. The algorithms can be generalized to a broader class of element shapes (e.g., simplices, similar to Ref. <cit.>). For the algorithms to be generalizable, the considered element must be able to define children (of the same type). Moreover, the element and its faces must be convex, so that the midpoint is guaranteed to be in the interior of the element. Note that this convexity requirement pertains to the shape of the untrimmed element, and not to the trimmed element.

§.§ Topology preservation

The spline-based segmentation procedure has been demonstrated to yield analysis-suitable domains for a wide range of test cases (see, e.g., Refs. <cit.>). An example from Ref. <cit.> is shown in Fig. <ref>. This example shows a carotid artery, obtained from a CT-scan. The scan data consists of 80 slices, separated by a distance of $400\,{\rm \mu m}$. Each slice image consists of $85\times 70$ voxels of size $300\times 300 {\rm \mu m}^2$.

Example of the spline-based segmentation procedure for a scan-based image of a carotid artery. (fig:topology_original) The original voxel data; (fig:topology_segmented) the spline segmentation based on the voxel mesh, resulting in a topologically erroneous domain; and (fig:topology_preserved) the spline segmentation after application of the topology-preservation algorithm.

When the spline-based segmentation procedure is performed using a B-spline level set function defined on the voxel grid, the result shown in Fig. <ref> is obtained. Although the smoothing characteristic of the technique is overall beneficial, in the sense that it leads to smooth boundaries, comparison to the original voxel data in Fig. <ref> shows that in this particular case the topology of the object is altered by the segmentation procedure. This can occur in cases where the features of the object to be described are not significantly larger in size than the voxels (i.e., the Nyquist criterion is not satisfied; see Section <ref>). In many cases, the altering of the topology of an object fundamentally changes the problem under consideration, and is therefore generally undesirable.

To avoid the occurrence of topological anomalies due to smoothing, a topology-preservation strategy has been developed in Ref. <cit.>. The developed strategy follows directly from the smoothing analysis presented in Section <ref>, which shows that features with a small relative length scale, $\hat{\ell}=\ell/h$, may be lost upon smoothing, cf. equation (<ref>). Hence, topological features may be lost when the mesh size on which the B-spline level set is constructed is relatively large compared to the voxel size. The pivotal idea of the strategy proposed in Ref. <cit.> is to detect topological anomalies by comparison of the segmented image (Fig. <ref>) with the original voxel data (Fig. <ref>) through a moving-window technique. In places where topological anomalies are detected, the mesh on which the smooth level set function is constructed is then refined locally (using THB-splines <cit.>). This locally increases the relative feature length scale, $\hat{\ell}$, such that the topology is restored (Fig. <ref>).


# Further generalizations of the parallelogram law

Antonio M. Oller-Marcén  Centro Universitario de la Defensa de Zaragoza
Ctra. Huesca s/n, 50090 Zaragoza, Spain<EMAIL_ADDRESS>
###### Abstract.

In recent work [2], a generalization of the parallelogram law in any dimension
$N\geq 2$ was given by considering the ratio of the quadratic mean of the
measures of the $N-1$-dimensional diagonals to the quadratic mean of the
measures of the faces of a parallelotope. In this paper, we provide a further
generalization considering not only $(N-1)$-dimensional diagonals and faces,
but the $k$-dimensional ones for every $1\leq k\leq N-1$.

##  1\. Introduction

If we consider the usual Euclidean space $(\mathbb{R}^{n},\|\cdot\|)$, the
well-known identity

(1) $\|a+b\|^{2}+\|a-b\|^{2}=2(\|a\|^{2}+\|b\|^{2})$

is called the parallelogram law.

This identity can be extended to higher dimensions in several ways. For
example, it is straightforward to see that

(2)
$\|a+b+c\|^{2}+\|a+b-c\|^{2}+\|a-b+c\|^{2}+\|a-b-c\|^{2}=4(\|a\|^{2}+\|b\|^{2}+\|c\|^{2})$

with subsequent analogue identities arising inductively. There are, in fact,
many works devoted to provide generalizations of (1) in many different
contexts [1, 3, 4]

Note that if we rewrite (1) as

(3)
$\frac{\|a+b\|^{2}+\|a-b\|^{2}}{2}=2\frac{(\|a\|^{2}+\|b\|^{2}+\|a\|^{2}+\|b\|^{2})}{4}$

it just means that in any parallelogram, the ratio of the quadratic mean of
the lengths of its diagonals to the quadratic mean of the lengths of its sides
equals $\sqrt{2}$. With this interpretation in mind, Alessandro Fonda [2] has
recently proved the following interesting generalization.

######  Theorem 1.

Given linearly independent vectors $a_{1},\dots,a_{N}\in\mathbb{R}^{n}$, it
holds that

$\displaystyle\sum_{i<j}\left(\left\|(a_{i}+a_{j})\wedge\bigwedge_{k\neq
i,j}a_{k}\right\|^{2}+\left\|(a_{i}-a_{j})\wedge\bigwedge_{k\neq
i,j}a_{k}\right\|^{2}\right)=$
$\displaystyle=(N-1)\sum_{k=1}^{N}2\left\|a_{1}\wedge\cdots\wedge\widehat{a_{k}}\wedge\cdots\wedge
a_{N}\right\|^{2}.$

In other words, for any $N$-dimensional parallelotope, the ratio of the
quadratic mean of the $(N-1)$-dimensional measures of its diagonals to the
quadratic mean of the $(N-1)$-dimensional measures of its faces is equal to
$\sqrt{2}$.

In this work we extend this result considering the faces of dimension $k$ for
every $1\leq k\leq N-1$ and a suitable definition of $k$-dimensional diagonal
of a parallelotope. Then, Theorem 1 will just be a particular case of our
result for $k=N-1$. Indeed, our result can be stated as follows.

######  Theorem 2.

Let us consider an $N$-dimensional parallelotope and let $1\leq k\leq N-1$.
The ratio of the quadratic mean of the $k$-dimensional measures of its
$k$-dimensional diagonals to the quadratic mean of the $k$-dimensional
measures of its $k$-dimensional faces is equal to $\sqrt{N-k+1}$.

In fact, our generalization goes in the line of the work [3] but considering
the definition of diagonal face given in [2].

##  2\. Notation and preliminaries

In this section we are going to introduce some notation and to present some
basic facts that will be useful in the sequel. Let us consider a parallelotope
$\mathcal{P}$ generated by a family of linearly independent vectors
$\mathcal{B}=\\{a_{1},a_{2},\dots,a_{N}\\}\subseteq\mathbb{R}^{n}$. This means
that

$\mathcal{P}=\left\\{\sum_{i=1}^{N}\alpha_{i}a_{i}:\alpha_{i}\in[0,1]\right\\}.$

Let us fix $1\leq k\leq N-1$. Then, given $k$ different vectors
$\mathcal{S}=\\{a_{i_{1}},\dots,a_{i_{k}}\\}\subseteq\mathcal{B}$, we can
consider the face generated by them:

$\mathcal{F}(\mathcal{S})=\left\\{\sum_{v\in\mathcal{S}}\alpha_{v}v:\alpha_{v}\in[0,1]\right\\}.$

This face can now be translated by one or more of the remaining vectors thus
obtaining a face

$\mathcal{F}^{I}(\mathcal{S})=\left\\{\sum_{v\in\mathcal{S}}\alpha_{v}a_{v}+\sum_{w\in\mathcal{B}\setminus\mathcal{S}}\alpha_{w}w\in\mathcal{P}:\alpha_{w}\in\\{0,1\\}\right\\},$

where $I=(\alpha_{v})_{v\not\in\mathcal{S}}\in\\{0,1\\}^{N-k}$. Since each
choice of a set $\mathcal{S}\subseteq\mathcal{B}$ and a vector
$I\in\\{0,1\\}^{N-k}$ leads to a different face and every face can be obtained
in this way, it follows the well-known result that $\mathcal{P}$ has exactly
$2^{N-k}\binom{N}{k}$ $k$-dimensional faces. Moreover, it is clear that all
the $2^{N-k}$ different faces $\mathcal{F}^{I}(\mathcal{S})$ are congruent to
the set generated by $\mathcal{S}$, $\mathcal{F}(\mathcal{S})$.

Now, we focus on the $k$-dimensional diagonals which will be defined following
the ideas in [2]. Let us consider $N-k+1$ different vectors
$\mathcal{T}=\\{a_{i_{1}},\dots,a_{i_{N-k+1}}\\}\subseteq\mathcal{B}$ and let
$\mathcal{T}=\mathcal{T}_{1}\cup\mathcal{T}_{2}$ be any partition. Then, the
following set

$\mathcal{D}(\mathcal{T}_{1},\mathcal{T}_{2})=\left\\{\alpha\sum_{v\in\mathcal{T}_{1}}v+(1-\alpha)\sum_{v\in\mathcal{T}_{2}}v+\sum_{w\in\mathcal{B}\setminus\mathcal{T}}\alpha_{w}w:\alpha,\alpha_{w}\in[0,1]\right\\}.$

is called the $k$-dimensional diagonal associated to
$(\mathcal{T},\mathcal{T}_{1},\mathcal{T}_{2})$. Clearly each choice of a set
$\mathcal{T}\subseteq\mathcal{B}$ and a partition of $\mathcal{T}$ leads to a
different diagonal. Thus, it readily follows that $\mathcal{P}$ has exactly
$2^{N-k}\binom{N}{N-k+1}$ different $k$-dimensional diagonals. Moreover, if we
define the vector

$V(\mathcal{T}_{1},\mathcal{T}_{2})=\sum_{v\in\mathcal{T}_{1}}v-\sum_{v\in\mathcal{T}_{2}}v,$

we have that

$\mathcal{D}(\mathcal{T}_{1},\mathcal{T}_{2})=\left\\{\alpha
V(\mathcal{T}_{1},\mathcal{T}_{2})+\sum_{v\in\mathcal{T}_{2}}v+\sum_{w\in\mathcal{B}\setminus\mathcal{T}}\alpha_{w}w:\alpha,\alpha_{w}\in[0,1]\right\\}$

and, consequently, it is clear that the diagonal
$\mathcal{D}(\mathcal{T}_{1},\mathcal{T}_{2})$ is just a translation of the
set generated by
$\\{V(\mathcal{T}_{1},\mathcal{T}_{2}),w:w\in\mathcal{B}\setminus\mathcal{T}\\}$
and, hence, it is congruent to it.

##  3\. Proof of Theorem 2

After introducing the notation and the main objects involved in thie work, we
are now in the condition to proof the main result of the paper.

Let $\mathcal{P}$ be a parallelotope generated by
$\mathcal{B}=\\{a_{1},a_{2},\dots,a_{N}\\}\subseteq\mathbb{R}^{n}$. We first
compute the quadratic mean of the $k$-dimensional measures of its
$k$-dimensional faces. To do so, we first note that, for every
$\mathcal{S}=\\{a_{i_{1}},\dots,a_{i_{k}}\\}\subseteq\mathcal{B}$, the
$k$-dimensional measure of the face $\mathcal{F}(\mathcal{S})$ is
$\|a_{i_{1}}\wedge\dots\wedge a_{i_{k}}\|$. In the previous section we have
seen that $\mathcal{P}$ has exactly $2^{N-k}\binom{N}{k}$ $k$-dimensional
faces and, moreover, that there are exactly $2^{N-k}$ copies of each face
$\mathcal{F}(\mathcal{S})$. Consequently, the quadratic mean of the
$k$-dimensional measures of the $k$-dimensional faces of $\mathcal{P}$ is:

(4) $\frac{\displaystyle 2^{N-k}\sum\|a_{i_{1}}\wedge\dots\wedge
a_{i_{k}}\|^{2}}{\displaystyle 2^{N-k}\binom{N}{k}}.$

Now we have to compute the quadratic mean of the $k$-dimensional measures of
the $k$-dimensional diagonals of $\mathcal{P}$. First of all, recall that
$\mathcal{P}$ has exactly $2^{N-k}\binom{N}{N-k+1}$ different $k$-dimensional
diagonals. Each of them is the translation of the set generated by
$\\{V(\mathcal{T}_{1},\mathcal{T}_{2}),w:w\in\mathcal{B}\setminus\mathcal{T}\\}$
for exactly one choice of $(\mathcal{T},\mathcal{T}_{1},\mathcal{T}_{2})$. The
$k$ dimensional measure of this latter set is
$\displaystyle\left\|V(\mathcal{T}_{1},\mathcal{T}_{2})\wedge\bigwedge_{w\in\mathcal{B}\setminus\mathcal{T}}w\right\|$.
Consequently, the quadratic mean of the $k$-dimensional measures of the
$k$-dimensional diagonals of $\mathcal{P}$ is:

(5)
$\frac{\displaystyle\sum_{\mathcal{T},\mathcal{T}_{1},\mathcal{T}_{2}}\left\|V(\mathcal{T}_{1},\mathcal{T}_{2})\wedge\bigwedge_{w\in\mathcal{B}\setminus\mathcal{T}}w\right\|^{2}}{\displaystyle
2^{N-k}\binom{N}{N-k+1}}.$

Now, using the bilinearity of the scalar product and taking into account the
definition of $V(\mathcal{T}_{1},\mathcal{T}_{2})$, it can be easily seen that
when we vary $(\mathcal{T},\mathcal{T}_{1},\mathcal{T}_{2})$, we get the term
$\|a_{i_{1}}\wedge\dots\wedge a_{i_{k}}\|^{2}$ exactly $2^{N-K}k$ times for
every possible choice of
$\\{a_{i_{1}},\dots,a_{i_{k}}\\}\subseteq\mathcal{B}$. This implies that the
quadratic mean of the $k$-dimensional measures of the $k$-dimensional
diagonals of $\mathcal{P}$ (5) can in fact be written as:

(6) $\frac{\displaystyle 2^{N-k}k\sum\|a_{i_{1}}\wedge\dots\wedge
a_{i_{k}}\|^{2}}{\displaystyle 2^{N-k}\binom{N}{N-k+1}}.$

Finally, in order to obtain Theorem 2 it is enough to divide (6) by (4):

$\frac{(\ref{qdf})}{(\ref{qf})}=\frac{k\binom{N}{k}}{\binom{N}{N-k+1}}=N-k+1.$

## References

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