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Superposition of layers of cubic lattice V. P. Grishukhin Central Economics and Mathematics Institute of the Russian Academy of Sciences, Moscow § 1. IntroductionThis paper, which continues [1], demonstrates on the example of the cubic lattice some ideas of the paper [1], in which most the required definitions can be found. The $n$-dimensional cube $Q^n$ is a simplest parallelohedron, and hence the method of superposition of layers becomes especially transparent and illustrative in the context of the cubic lattice. In addition, this example illustrates, as a virtuous glass, many important properties of parallelohedra, lattices, and quadratic forms. Recall that a parallelohedron is a convex polytope whose face-to-face translates tile the entire space. A parallelohedron of dimension $n$ is called primitive if each of its vertices is common to $n+1$ parallelohedra from the tiling of the space by this parallelohedron. The centres of the parallelohedra of this tiling (or a partition) $\mathcal T$ form a point lattice $L$. If two parallelohedra $P,P'$ of a partition $\mathcal T$ have non-empty intersection, then the intersection $P\cap P'=G$ is a face of these two parallelohedra (this face is known as a contact face). A contact face of maximal dimension (that is, a face of codimension 1) is a facet. A contact vector is a vector of the lattice $L$ which connects centres of parallelohedra with common contact face. Dolbilin (see, for example, [2]) was the first to consider and study contact faces. A particular case of a parallelohedron is the classical Dirichlet–Voronoi cell (a DV-cell) of some point $l$ of the lattice $L$. This cell is the closure of all points of the space which are closer to $l$ (with respect to the Euclidean metric $l^2$) than to other points of $L$. The DV-cell of the cubic lattice $Z^n$ is the unit cube $Q^n$. If a partition $\mathcal T$ is formed by classical DV-cells, then the contact vectors $l$ have the smallest $l^2$-norm in their parity class (the quotient group $L/2L$). A parity class is called prime if it contains a single (up to sign) contact vector, which is called a facet vector in this case. The facet vector connects the centres of two DV-cells with common facet, and, since the DV-cells are classical, the above facet vector is orthogonal to the corresponding facet. A compound class is a non-prime parity class. Contact vectors play an important role in the present paper. Delaunay was the first to introduce and employ the method of superposition of layers in his paper [3] on enumeration of types of four-dimensional parallelohedra. For the method of superposition of layers, see, for example, [4], [5], [1], and other studies (see also § 3 of the present paper). The lattice $L^{n+1}$ obtained by superposition of layers isomorphic to the $n$-dimensional lattice $L$ is uniquely defined by the lattice $L$ and two parameters: the distance (height) $h$ between the layers in the vertical direction along the unit vector $e$ and the vector $v$ of shift in the horizontal direction of one layer with respect to the neighbouring layer. In our case, the $n$-dimensional cubic lattice $L=Z^n$, the shift vector $v$ is a vector with end-point at any vertex of the cube $Q^n$, and the result does not depend on the chosen vertex. Hence the main parameter is the height $h$. Let $L^{n+1}_Z(h)$ be the lattice obtained by superposition of layers of the cubical $n$-dimensional lattice $Z^n$, and $P^{n+1}(h)$ be its DV-cell. In the present paper, we describe in detail the behaviour of the lattice $L_Z^{n+1}(h)$, its DV-cell $P^{n+1}(h)$, and its metric form $f^{n+1}_h(x)=\sum_{i\in N}x_i^2+4h^2x_{n+1}^2$ as a function of the parameter $h$, where $N=\{1,2,\dots ,n\}$ is a set of indexes. With the family of DV-cells $P^{n+1}(h)$ one can conveniently associate the line of metric forms $f^{n+1}_h$, which intersects the cone ${\mathcal A}_{n+1}$ of non-negative quadratic forms from one boundary of the cone to the other boundary. Voronoi [6] defined a partition of the cone ${\mathcal A}_n$ of dimension $n(n+1)/2$ into $L$-domains, each of which is a polyhedral cone of dimension from 1 to $n(n+1)/2$. To quadratic forms of a single $L$-domain there correspond lattices whose DV-cells are of the same combinatorial structure. The forms lying in one-dimensional $L$-domains are called edge-forms. The corresponding DV-cells are called mainstay cells in [7]. If $h$ increases from 0 to $\infty$, the form $f^{n+1}_h$ intersects the cone ${\mathcal A}_{n+1}$ and passes through $n-\lfloor n/2\rfloor$ edge-forms. One of the main results of the paper is the proof that the DV-cell $P^{n+1}(h)$ is the intersection of the generalized cube $Q^{n+1}(h)$ and the generalized ortahedron $O^{n+1}(h)$. By definition, the ortahedron (cross-polytope) $O^n$ of dimension $n$ is a regular polytope dual to the cube $Q^n$. So, the ortahedron is the convex hull of centres of all $(n-1)$-dimensional faces of the cube $Q^n$. In the dimension $n= 3$, the ortahedron is the octahedron. The term ortahedron was introduced by E. P. Baranovskii and S. S. Ryshkov (see, for example, § 3.1 of [5] and § 2$^\circ$ of [7]). So, we have where $Q^{n+1}(h)$ and $O^{n+1}(h)$ are, respectively, the dilated (along the vector $e$) $(n+1)$-dimensional cube $Q^{n+1}$ and the $(n+1)$-dimensional polytope $\mu_{n+1}O^{n+1}$, which is homothetic to the $(n+1)$-dimensional ortahedron $O^{n+1}$ with homothety coefficient $\mu_{n+1}=(n+1)/2$. Note that $Q^{n+1}(0)=Q^n$ and $O^{n+1}(0)=\mu_nO^n$.Lemma 1 of [7] asserts that the intersection $Q^n\cap\mu_nO^n$ is the DV-cell $P(D_n^*)$, where $D_n^*$ is the dual lattice to the root lattice $D_n$. This is an additional evidence for the result of Proposition 5 (see § 6), which asserts that $P^{n+1}(0)=P(D_n^*)$ and $P^{n+1}(1/2)=P(D_{n+1}^*)$. For further purposes, it is worth recalling that, in the orthonormal basis ${\mathcal B}^n$, the lattice $D_n^*$ consists of vectors with integer coordinates or of vectors with half-integer coordinates. In the present paper, the unit cube is the DV-cell of the cubic lattice $Z^n$, and hence the facet vectors of the cube $Q^n$ are unit vectors $\pm b_i\in {\mathcal B}^n$ for all $i\in N$, where the inner product $\langle b_i,b_j\rangle$ is 0 if $i\not=j$, and $b_i^2=1$ for all $i\in N$. The facet vectors of the ortahedron $\mu_nO^n$ are as follows: $v(S)=b(S)-b(N)/2$, where $b(S)=\sum_{i\in S}b_i$, for all subsets $S\subseteq N$. The vectors $v(S)$ define the facets, which split a simplex from each of the $2^n$ coordinate angles; the union of these simplexes is the ortahedron $\mu_nO^n$. In comparison with $Q^n$, the cube $Q^{n+1}(h)$ of dimension $n+1$ in intersection (1.1) has additional facet vectors $\pm b_{n+1}=\pm 2he$, which, for $h<h_n$, where $h_n^2=n/4$, define the horizontal facets of the intersection $P^{n+1}$. In intersection (1.1), the elongated ortahedron $O^{n+1}(h)$ has facets $\pm F(S,h)$ with facet vectors $\pm(v(S)+he)$ for all $S\subseteq N$; these facets depend on $h$. For small $h$, the cube $Q^{n+1}(h)$ is nearly flat, and the facets $\pm F(S,h)$ of the ortahedron $O^{n+1}(h)$ are nearly vertical. With increasing $h$, the cube $Q^{n+1}(h)$ elongates, and the ortahedron $O^{n+1}(h)$ is contracted along the vertical vector $e$. Let $H_{\pm 1/4}$ be the hyperplanes orthogonal to the central axis, which is spanned by the vector $e$ and which intersects this axis at distance $\pm he/2$ from the origin. With increasing $h$, the facets $\pm F(S,h)$ of the ortahedron are inclined to the central axis, so that their intersections with the hyperplanes $H_{\pm 1/4}$ remain intact. For $h=h_n$, the upper and lower vertices of the ortahedron $O^{n+1}(h)$ lie on the upper and lower faces of the cube $Q^{n+1}(h_n)$, and these faces become vertices of the DV-cell $P^{n+1}(h_n)$. For $h>h_n$, the DV-cell $P^{n+1}(h)$ becomes a zonotope (see the next section). Now the DV-cells $P^{n+1}(h)$ for $n=1$ and $n=2$ can be easily described. Let $n=1$. Then $P^2(0)$ and $Q^2(0)$ are closed intervals, and $h_n^2=h_1^2=1/4$. Hence, if $0<h<h_1=1/2$, then $Q^2(h)$ is a rectangle, and the ortahedron $O^2(h)$ is an elongated rhomb, whose longer diagonal is a part of the central vertical axis. The intersection $P^2(h)=Q^2(h)\cap O^2(h)$ is a hexagon with horizontal upper and lower edges, which are parts of the upper and lower edges of the rectangle $Q^2(h)$. For $h=h_1=1/2$, the DV-cell $P^2(1/2)$ becomes the square $Q^2(1/2)=O^2(1/2)$, since, for $h=1/2$, the basis is orthonormal. If $h>h_1=1/2$, then $P^2(h)$ becomes a hexagon again, but now with lateral vertical edges. Let $n=2$. Then $P^3(0)$ and $Q^3(0)$ are squares, and $h_n^2=n/4=h_2^2=1/2$. Hence, if $0<h^2<h^2_2=1/2$, then the ortahedron $O^3(h)$ is an elongated octahedron, whose longer diagonal is a part of the central axis, and the intersection $P^3(h)=Q^3(h)\cap O^3(h)\cong P(D_3^*)=P^3(1/2)$ is the truncated octahedron (which is the only type of the three-dimensional primitive parallelohedron). For $h^2=h_2^2=1/2$, the upper and lower square faces of the truncated dodecahedron shrink to vertices, and the DV-cell $P^3(h_2)$ becomes a dodecahedron. If $h>h_2$, then $P^3(h)$ becomes an elongated dodecahedron. § 2. The structure of the paperIn § 3, we describe the method of superposition of layers of the $n$-dimensional lattice $L$, which is the method of construction of the lattice $L^{n+1}(h)$ of plus one dimension. Table 1 of § 4 shows the set ${\mathcal L}(h)$ of all vectors of lattices $L_Z^{n+1}(h)$ which are contact vectors for some $h$. Below, an important role is played by the parity classes $C(S)$ of contact vectors, which depend on sets $S\subseteq N$. The parity class $C(S)$ consists of contact vectors of contact faces of the elongated cube $Q^{n+1}(h)$. To each set $S\subseteq N$ of cardinality $s=|S|$, there correspond $2^s$ faces independent of $n$, and $2^{n-s}$ faces depending on $h$. Correspondingly, the set of contact vectors of the parity class $C(S)$ is split into two subsets ${\mathcal L}_h(S)$ and ${\mathcal L}_0(S)$ of contact vectors which depend or do not depend on $h$. If the cardinality $|S|=s$ of $S\subseteq N$ is smaller than $n/2$, then, for all $h$, the contact vectors are the only vectors of the set ${\mathcal L}_0(S)$. But, if $s\geqslant n/2$, then, as $h$ increases from 0 to infinity, first, the contact vectors are only vectors from the set ${\mathcal L}_h(S)$, and second, the contact vectors are only vectors from the set ${\mathcal L}_0(S)$. Thus modification takes place for the critical value $h=h_s=(2s-n)/4$, when contact vectors are vectors from both sets. In § 5, we define the space $M_C({\mathcal L})$ of symmetric matrices generated by the set of contact vectors ${\mathcal L}$ of an arbitrary lattice $L$. This space is described by linear equalities of $l^2$-norms of non-collinear contact vectors $l=\sum_{i\in N}e_ix_i\in{\mathcal L}$ of a single compound parity class. The norms $l^2=\sum_{ij}a_{ij}x_ix_j$ are linear forms of the coefficients $a_{ij}=\langle e_i,e_j\rangle$ of the Gram matrix of the basis ${\mathcal E}=\{e_i\colon i\in N\}$ of the $n$-dimensional lattice $L\ni l$. The subscript “C” is indicative of C-types of lattices introduced by S. S. Ryshkov and E. P. Baranovskii in the book [8], where contact vectors are called adjacency vectors. Hence, the lattices with the same set of contact vectors are of the same C-type (adjacency type). In § 5, we also define a C-extremal set ${\mathcal L}$ of contact vectors — this is a set for which the space $M_C({\mathcal L})$ is one-dimensional. In other words, a C-extremal set ${\mathcal L}$ defines uniquely (up to a factor) the Gram matrix of a basis of the corresponding lattice. It is shown that the sets ${\mathcal L}(h_s)$ of contact vectors of the lattice $L^{n+1}_Z(h)$ for the critical values of the height $h=h_s$ are C-extremal for integer $s$ in the range $n/2\leqslant s\leqslant n-1$. In our case, the beginning of the line
$$
\begin{equation*}
\bigl\{f\in{\mathcal A}_{n+1}\colon f=f^{n+1}_h,\ 0\leqslant h\leqslant\infty\bigr\}
\end{equation*}
\notag
$$
of the forms, which pierce the cone ${\mathcal A}_{n+1}$, is the form $f^{n+1}_0=f(D_n^*)$ of the $n$-dimensional lattice $D_n^*$. The form $f(D_n^*)$ is an edge-form only for even $n$ (cf. Theorem 1 in [7], which asserts that $f(D_n^*)$ is an edge-form for all $n$). The lattice $D_n^*$ appears two times in the family of lattices $L_Z^{n+1}(h)$ obtained by superposition of the layers of the cubic lattice $Z^n$. Namely, $L_Z^{n+1}(0)=D_n^*$ is the lattice generated by the original lattice $Z^n$ and the vector $b(N)/2$. Next, for $h=1/2$, this lattice appears again as the lattice $L_Z^{n+1}(1/2)=D^*_{n+1}$. For each $n$, only one of these lattices is an edge-lattice. In § 6, we consider the DV-cell $P^{n+1}(h)$ of the lattice $L_Z^{n+1}(h)$. We will also prove equality (1.1). For the critical values of $h=h_s$, where $s$ is an integer number in the interval $n/2\leqslant s\leqslant n-1$, in the family of DV-cells $P^{n+1}(h)$ there appear mainstay DV-cells $P^{n+1}(h_s)$ corresponding to edge-forms $f^{n+1}_{h_s}$. If $s=n$, then to the critical value of $h_n^2=n/4$ there corresponds the form $f^{n+1}_{n/4}(x)=\sum_{i\in N}x_i^2+nx_{n+1}^2$, which is not an edge-form. The facets $F(S,h)$ of the ortahedron $O^{n+1}(h)$ intersect at the points $\pm v^0(h)$, which lie on the central axis. For $h=h_n$, the points $\pm v^0(h_n)=\pm b_{n+1}/2$ lie on the horizontal facets $F(\pm b_{n+1})$ of the cube $Q^{n+1}(h_n)$, and these facets become the contact vertices $\pm v^0(h_n)$. Correspondingly, the vectors $\pm b_{n+1}$ cease to be facet vectors, and become contact vectors, and the DV-cell $P^{n+1}(h_n)$ becomes a zonotope, which is considered in § 7. The zonotope $Z(U)$, which is a DV-cell, is the Minkowskii sum of segments of straight lines spanned by the vectors of a unimodular set $U$. A set of vectors $U\subset{\mathbb R}^n$ is called unimodular if all its vectors are expressible, as integer combinations, in terms of vectors of any of its basis subset, that is, of a subset consisting of $n$ linearly independent vectors. The root system ${\mathbb A}_n$ has greatest cardinality among all $n$-dimensional unimodular sets. In our case, $U\subset{\mathbb A}_{n+1}$ for all $h\geqslant h_n$. The structure of the zonotope $Z(U)$ changes as $h$ passes through the point $h_n$. We show that its facets $F(S,h)$ are $n$-dimensional parallelepipeds, and the lateral facets are the DV-cells $P^n(h)$. In the last § 8, we describe in detail the four-dimensional DV-cells $P^4(h)$. Three values of $h$ are singled out: $h=0$, and two critical values $h_2=1/2$ and $h_3^2=3/4$. The DV-cell $P^4(0)$ is the 3-dimensional zonotope $Z({\mathbb A}_3)=Q^3\cap O^3$, which is the truncated octahedron. The DV-cell $P^4(1/2)=P(D_4^*)\cong P(D_4)$ is a mainstay parallelohedron, which is a regular polytope known as the 24-cell. The DV-cells $P^4(h_3)=Z(U^4_0)$ and $P^4(h)=Z(U^4_h)$ are zonotopes, where $U^4_0$ and $U^4_h$ for $h>h_3$ are unimodular subsets $U\subset{\mathbb A}_4$. For the remaining $h$ in the ranges $0<h<1/2$ and $1/4<h^2<3/4$, the DV-cells $P^4(h)$ are sums of the mainstay DV-cell $P(D_4)$ with some zonotopes. § 3. Superposition of layersSuperposition of layers of some $n$-dimensional lattice $L$ is a method for producing a new lattice $L^{n+1}(h)$ of dimension $n+1$. In addition to the lattice $L$ itself, superposition of its layers involves two parameters: the distance (height) $h$ between layers in the vertical direction along the unit vector $e$ and the vector $v$ of shift in the horizontal direction of one layer with respect to neighbouring one. Of special interest is the superposition of layers of the lattice $L$, where as a shift vector one takes a vector with end-point at a vertex of the parallelohedron $P$, for which $L$ is the translation lattice (recall that the origin 0 is the centre of $P$). In this case, the result depends not on the chosen vertex, but rather on its class $V'$ of translation equivalence of vertices. Two vertices are equivalent if one of them can be obtained from a different one by translation by a vector of the lattice $L$. For example, in the cube $Q^n$, all its vertices compose a single equivalence class. So, the lattice $L^{n+1}(h)$ is generated by the original lattice $L$, which forms the zero layer, and any of the vectors $v+he$ of the first layer, where the vertex $v$ is identified with the vector for which $v$ is its end-point. If $h=0$, then the lattice $L^{n+1}(0)$ is $n$-dimensional; this lattice is known as the centering of the original lattice $L$ by all vectors $v\in V'$. B. N. Delaunay was the first to apply the method of superposition of layers. In this way, he noted that, for sufficiently large height $h$ (more precisely, if $h$ exceeds the radius $R(P)$ of the sphere described about the DV-cells $P$ of the original lattice), then the DV-cell $P^{n+1}(h)$ is a lifting of the DV-cell $P$. This means, in particular, that the original DV-cell $P$ is the projection along the vector $e$ of the DV-cell $P^{n+1}(h)$ if $h\geqslant R(P)$. Under this projection, the original DV-cell $P$ is split into pieces which are intersections of $P$ with the DV-cells of the original partition of the space into copies of $P$ which is translated by the shift vector. A partition of $P$ in pieces is called a sketch. The paper [4] shows 51 such sketches. In the superposition of layers of the cubic lattice $Z^n$ considered in the present paper, the DV-cell $P$ is the unit cube $Q^n$. The squared radius of the circumsphere is $R^2(Q^n)=n/4$, that is, it is equal to the squared last critical value $h_n$. Hence, for $h^2\geqslant h_n^2=n/4$, the DV-cell $P^{n+1}(h)$ is a lifting of the cube $Q^n$. This sketch of the cube is fairly simple. This sketch is a partition of the cube $Q^n$ into $2^n$ smaller cubes with a common vertex at the centre of the original cube. In § 6, we show that if $h\geqslant h_n$, then the DV-cell $P^{n+1}(h)$ has facets of two types: $2^{n+1}$ facets $\pm F(S,h)$ of the upper and lower caps for all subsets $S\subseteq N$ and $2n$ lateral facets $F(\pm b_i)$ of the equator for all $i\in N$. The equator of the DV-cell $P^{n+1}(h)$ consists of the faces of codimension 1 and 2 which are projected into facets of the projection. Each facet of the caps is projected in a small cube of the sketch, and hence the facet itself is an $n$-dimensional parallelepiped. In the dimension 3, the zonotope DV-cell $P^3(h_3)$ is a dodecahedron, for which each face is a square. § 4. The lattice $L_Z^{n+1}(h)$ obtained by superposition of layers of the lattice $Z^n$Let ${\mathcal B}^n=\{b_i\colon i\in N\}$ be an orthonormal basis, that is, $\langle b_i,b_j\rangle=\delta_{ij}$, where $\delta_{ij}$ is the Kronecker delta. For each $S\subseteq N$, we define $b(S)=\sum_{i\in S}b_i$. The basis ${\mathcal B}^n$ generates, via integer combinations, the cubic lattice $Z^n$ whose DV-cell is the unit cube
$$
\begin{equation}
Q^n=\biggl\{x\in\mathbb{R}^n\colon -\frac{1}{2}\leqslant\langle x,b_i\rangle\leqslant\frac{1}{2}\text{ for all }i\in N\biggr\}.
\end{equation}
\tag{4.1}
$$
The cube is a simplest parallelohedron. All its faces are contact faces. The corresponding contact vectors have the form $b(S)-2b(T)$ for all $T\subseteq S$; they lie in the parity class $C(S)$, which is defined uniquely by the set $S$. All vertices of the cube compose an equivalence class. Hence the lattice $L_Z^{n+1}(h)$ obtained by superposition of layers of the cubic lattice $Z^n$ with the use of a vertex of the cube $Q^n$ as a shift vector is uniquely determined by the height $h$. The shift vectors coincide with the vertices $v(S)=b(S)-b(N)/2$ of the cube $Q^n$, and hence $L_Z^{n+1}(0)=D_n^*$ is a centering of the lattice $Z^n$ by the vectors $v(S)$, that is, it is generated by the lattice $Z^n$ and the vector $v(S)+he$ for any $S\subseteq N$. We extend the basis ${\mathcal B}^n$ with the vector $b_{n+1}=2he$ of norm $b_{n+1}^2=4h^2$ to the basis ${\mathcal B}^{n+1}(h)$. We set $N'=N\cup\{n+1\}$. Now the vectors $v(S)\pm he$ have the form $v'(S)=b(S)-b(N')/2$, where $S\subseteq N'$. The Gram matrix of the new basis is the diagonal $(n+1)\times(n+1)$ matrix $A(h)$ in which the first $n$ diagonal entries are 1’s, and the last $(n+1)$st entry is $4h^2$. For $h=1/2$, the basis ${\mathcal B}^{n+1}(1/2)$ becomes an orthonormal basis ${\mathcal B}^{n+1}$, and, similarly to the case of $h=0$, we have $L_Z^{n+1}(1/2)=D^*_{n+1}$. All (up to sign) $2^n$ vectors $v'(S)=b(S)-\frac{1}{2}b(N')$ for all subsets $S\subseteq N'$ are contact vectors of the new parity classes $C(S,h)$ with minimal norm $v'(S)^2=n/4+h^2$. The class $C(S,h)$ has (up to sign) only one contact vector $v'(S)$. This means that all $2^n$ parity classes $C(S,h)$ are prime, and the vectors $v'(S)$ are facet vectors for all $S\subseteq N'$ and $v'({\overline S})=-v'(S)$. Here, for $S\subseteq N'$, we set $\overline S=N'\setminus S$. Table 1.The contact vectors of the lattice $L^{n+1}(h)$
The sets of contact vectors of old parity classes $C(S)$, where $S\not=\varnothing$, are augmented with new vectors of the form $b({\overline S})-2b(T')$ for all $T'\subseteq{\overline S}\subset N'$, which are contact vectors for some $h$. That these vectors lie in the parity class $C(S)$ follows from the equality
$$
\begin{equation*}
(b(S)-2b(T))+(b(\overline S)-2b(T'))=2\biggl(\frac{1}{2}b(N')-b(X)\biggr)=-2v'(X),
\end{equation*}
\notag
$$
where $T\cap T'=\varnothing$, because $T\subseteq S$, $T'\subseteq\overline S$, $X=T\cup T'$ and $v'(X)=b(X)-\frac{1}{2}b(N')$. We split the set of contact vectors of the parity class $C(S)$, where $S\subseteq N$, into the following two subsets
$$
\begin{equation*}
\begin{aligned} \, \mathcal{L}_0(S) &=\{b(S)-2b(T),\, T\subseteq S\}, \\ \mathcal{L}_h(S) &=\{ b(N\setminus S)-2b(T)\pm 2he,\, T\subseteq{N\setminus S}\}. \end{aligned}
\end{equation*}
\notag
$$
The norms of vectors of these subsets are $|S|=s$ and $n-s+4h^2$, respectively. Hence, for all $h\geqslant 0$, for $s<n/2$, the contact vectors are vectors from the set ${\mathcal L}_0(S)$. For $s\geqslant n/2$, the contact vectors are vectors from the set ${\mathcal L}_h(S)$ if the inequality $n-s+4h^2<s$ holds, that is, if $4h^2<2s-n$. This inequality becomes an equality with increasing $h$; in this case, the contact vectors are vectors from both sets ${\mathcal L}_0(S)$ and ${\mathcal L}_h(S)$. A value of $h^2_s$ for which the above inequality becomes an equality will be referred to as a critical value. Setting $m=\lfloor n/2\rfloor$, we have
$$
\begin{equation}
h_s^2=\frac{2s-n}4,\quad\text{where }\ s \ \text{ is an integer such that }\ m\leqslant s\leqslant n,\ \text{ and } \ h_n^2=\frac{n}4.
\end{equation}
\tag{4.2}
$$
So, let $s\geqslant m$. In this case, the set of contact vectors of parity class $C(S)$ is
$$
\begin{equation}
\begin{cases} \mathcal{L}_h(S) &\text{if }h<h_s, \\ \mathcal{L}_h(S)\cup\mathcal{L}_0(S) &\text{if }h=h_s, \\ \mathcal{L}_0(S) &\text{if }h>h_s. \end{cases}
\end{equation}
\tag{4.3}
$$
Proposition 1 describes the intervals of variation of $h$ in which the set of contact vectors does not change. Proposition 1. Let $n=2m$ and $n=2m-1$, for even and odd $n$, respectively. Then the range of variation of the height $h$ splits into $2m+2$ intervals of the form If $n=2m$ is even, then the equality $h=h_m$ becomes $h=0$, and the interval $0<h<h_m$ is empty. In addition, $P^{n+1}(h_m)=P(D^*_{2m})$ is of dimension $n$ if $n$ is even, and is of dimension $n+1$ if $n=2m-1$ is odd. Proof. The cases of even and odd $n$ are different. If $n=2m$ is even, then $s=n/2= m$ is the smallest value for $s$, and, for this $s$, we have $h_m=0$. In this case, it is clear that the interval $0<h<h_m$ is empty, and $P^{n+1}(h_m)=P^{n+1}(0)=P(D_n^*)$ is of dimension $n=2m$.
If $n=2m-1$ is odd, then the smallest value is also $s=m$. But now, for $s=m$, we have $h_m^2=1/4\ne 0$, that is, $h_m=1/2$. In this case, $P^{n+1}(h_m)= P^{n+1}(1/2)= P(D^*_{n+1})$ has full dimension $n+1=2m$. It is easily seen that the number of pairs of intervals of the form $(h_s$, $h_s<h< h_{s+1})$ for all $s$, $m\leqslant s\leqslant n-1$, is $m$ if $n=2m$ is even, and is $m-1$ if $n=2m-1$ is odd. A pair of intervals $(h=0$, $0<h< h_m)$ exists only for an odd $n=2m-1$, and hence, for odd $n$, in addition to pairs of intervals $(h_s,\text{ }h_s<h<h_{s+1})$, there appears this additional pair. By adding the last pair of intervals $h^2=n/4$ and $h^2>n/4$, we get in total $m+1$ pairs of interval. $\Box$ § 5. The spaces $M_C(h)$Let contact vectors $l\in{\mathcal L}$ be defined in some basis ${\mathcal E}=\{e_i\colon i\in N\}$ (not necessarily in the basis of the lattice $L$). Let $a_{ij}$ be the coefficients of the Gram matrix of the basis ${\mathcal E}$. If $l=\sum_{i\in N}l_ie_i$ is an expansion of $l$ in vectors $e_i$ of the basis ${\mathcal E}$, then the norm
$$
\begin{equation*}
l^2=\sum_{i,j\in N}l_il_j\langle e_i,e_j\rangle=\sum_{i,j\in N}l_il_ja_{ij}=\langle l,Al\rangle
\end{equation*}
\notag
$$
is a linear form of the coefficients $a_{ij}$ of the Gram matrix $A$ of the basis ${\mathcal E}$. The set ${\mathcal L}$ of contact vectors generates the subspace $M_C({\mathcal L})$ of the space ${\mathbb R}^{n(n+1)/2}$ of all symmetric $n\times n$ matrices. Comparing the $l^2$-norms of non-collinear contact vectors $l$ of the same compound parity class, we get a homogeneous equation for the unknown coefficients $a_{ij}$ of the Gram matrix $A$. Each such equation defines a hyperplane in the space ${\mathbb R}^{n(n+1)/2}$. The intersection of all such hyperplanes defines the subspace $M_C({\mathcal L})$ of the space ${\mathbb R}^{n(n+1)/2}$. We say that a set ${\mathcal L}$ of contact vectors is $C$-extremal if the space $M_C({\mathcal L})$ is one-dimensional. It can be shown that the one-dimensional spaces $M_C({\mathcal L})$ are simultaneously one-dimensional $L$-domains. Hence the quadratic form $f(x)=\langle x,Ax\rangle$ whose matrix $A$ lies in the one-dimensional space $M_C({\mathcal L})$ is an edge form, and the corresponding DV-cell is a mainstay parallelohedron. In the basis ${\mathcal B}^{n+1}(h)$, the set of vectors of the lattice $L_Z^{n+1}(h)$ which are contact for some values of $h$ including the facet vectors is given in Table 1. We let ${\mathcal L}(h)$ denote the set of contact vectors of the lattice $L_Z^{n+1}(h)$. Let $M_C(h)=M_C({\mathcal L}(h))$, where the space $M_C({\mathcal L})$ is defined above. Below, we will show that if $h=h_s$, where $m\leqslant s\leqslant n-1$, then the set ${\mathcal L}(h)$ is C-extremal, and the DV-cells $P^{n+1}(h_s)$ are mainstay ones. If $n=2$, then the DV-cells $P^3(h)$ are three-dimensional. In addition, all 3-dimensional parallelohedra are zonotopes which may fail to be mainstay. So, in what follows, we assume that $n\geqslant 3$. Proposition 2. Let $n\geqslant 3$ and $m=\lfloor n/2\rfloor$. Then the set of contact vectors ${\mathcal L}(h)$ is C-extremal if $h=h_s$ for any integer $s$ from the interval $m\leqslant s\leqslant n-1$. Proof. Assume that the contact vectors in Table 1 are given not in the basis ${\mathcal B}^{n+1}(h)$, but in some basis ${\mathcal E}=\{e_i\colon i\in N'\}$ with the same coordinates. Let us show that, for $h=h_s$, such contact vectors define uniquely, up to a common factor, the coefficients $a_{ij}$ of the Gram matrix of the basis ${\mathcal E}$.
Recall that, for all $h\geqslant 0$, if $s<n/2$, then the contact vectors are vectors from the set ${\mathcal L}_0(S)$, where $|S|=s$. Hence if $n\geqslant 5$, that is, $m\geqslant 3$, then $n/2>2$, and the vectors from the set ${\mathcal L}_0(S)$, where $|S|=2$, are contact vectors. If $m=2$, that is, $n=3$ or $n=4$, then $h_m=h_2$ is a critical value. Hence by (4.3), the vectors of the set ${\mathcal L}_0(S)$, where $|S|=2$, are also contact vectors. So, for all critical values of $h=h_s$, where $m\leqslant s\leqslant n-1$, the compound parity class $C(\{ij\})$, where $i\not=j\in N$, contains the contact vectors $e_i\pm e_j$. The equality of the norms of the vectors implies that $\langle e_i,e_j\rangle=0$. This equality holds for pairs of non-equal indexes $i,j\in N$, and hence the vectors of the basis $\mathcal E$ are pairwise orthogonal for these pairs. According to (4.3), if $h\leqslant h_{n-1}$ and $|S|=n-1$, then the set of contact vectors of parity class $C(S)$, where $|S|=n-1$, of the lattice $L_Z^{n+1}(h)$ contains the set ${\mathcal L}_h(S)$. Let $S=N\setminus\{i\}$. Then $e(N'\setminus S)=e_i+e_{n+1}$ and ${\mathcal L}_h(S)=\{\pm e_i\pm e_{n+1}\}$. Hence $(e_i+e_{n+1})^2=(e_i-e_{n+1})^2$, that is, $\langle e_i,e_{n+1}\rangle=0$. This equality holds for all $i\in N$ and for all $h^2\leqslant h^2_{n-1}$. Consider the sets $\{e(S)-2e(T)$, $T\subseteq S\}$, where $S\subseteq N$, $m\leqslant |S|\leqslant n-1$, and the cardinality $|S|=s$ is fixed. If $h=h_s$, then by (4.3), for $T\subseteq S$ and $T'\subseteq N\setminus S$, the vectors $e(S)-2e(T)$ and $e(N\setminus S)-2e(T')+e_{n+1}$ are contact vectors. According to the above, $\langle e_i,e_j\rangle=0$, where $i\ne j\in N'$. Using these equalities and comparing the norms of the above vectors $e(S)-2e(T)$ and $e(N'\setminus S)-2e(T')$, we find that
$$
\begin{equation}
\sum_{i\in S}e_i^2=\sum_{i\in N\setminus S}e_i^2+e^2_{n+1}.
\end{equation}
\tag{5.1}
$$
Since $s\leqslant n-1$, we have $S\ne N$. Hence there exist $i\in S$, $j\in N\setminus S$ and $S'=S\setminus\{i\}\cup\{j\}$ such that $|S'|=s$. Consider the equality of norms (5.1), where $S$ is replaced by $S'$. Subtracting the equality for $S'$ from that for $S$, we get the equation $e_i^2-e_j^2=e_j^2-e_i^2$, that is, $e_i^2=e_j^2$. Since, for each pair $i,j\in N$, there exists a pair of corresponding sets $S$ and $S'$, the norms are equal to, say, $e_i^2=\alpha$, for all $i\in N$. Now equality (5.1) assumes the form $s\alpha=(n-s)\alpha+e^2_{n+1}$. Hence $e^2_{n+1}=(2s-n)\alpha$. We have $2s-n=4h_s^2$, and hence, for $h=h_s$, we obtain that $e^2_{n+1}=4h_s^2\alpha$. So, if $h=h_s$, then the coefficients $a_{ij}$ of the Gram matrix are defined uniquely up to a positive factor.$\Box$ § 6. The DV-cells $P^{n+1}(h)$By definition, the classical Dirichlet–Voronoi cell (DV-cell) $P(L)$ of an arbitrary $n$-dimensional lattice $L$ can be described by inequalities $x^2\leqslant(x-l)^2$, that is, it has the description by linear inequalities
$$
\begin{equation}
P(L)=\biggl\{x\in\mathbb{R}^n\colon \langle x,l\rangle\leqslant\frac{1}{2}\, l^2\text{ for all } l\in\mathcal{F}\biggr\},
\end{equation}
\tag{6.1}
$$
where ${\mathcal F}$ is the set of facet vectors of the lattice $L$. Recall that $P^{n+1}(h)$ is, by definition, the classical DV-cell of the lattice $L_Z^{n+1}(h)$. The C-extremal sets of contact vectors define the mainstay parallelohedra, and so Proposition 2 has the following corollary. Corollary 1. The DV-cells $P^{n+1}(h_s)$ are mainstay parallelohedra for all integer $s$ in the interval $n/2\leqslant s\leqslant n-1$. In the basis ${\mathcal B}^{n+1}(h)$, the facet vectors belong to the following set
$$
\begin{equation}
\mathcal{F}(h)=\{\pm b_i,\, i\in N,\, \pm b_{n+1}\}\cup \biggl\{b(S)-\frac{1}{2}(b(N)\pm b_{n+1}), \, S\subseteq N\biggr\},
\end{equation}
\tag{6.2}
$$
where, we recall, $b_i^2=1$ for $i\in N$ and $b_{n+1}^2=4h^2$. The facet vectors of the first set in (6.2) describe the following $(n+1)$-dimensional cube expanded by the vector $e$:
$$
\begin{equation}
Q^{n+1}(h)=\biggl\{x\in\mathbb{R}^{n+1}\colon -\frac{1}{2}\leqslant x_i\leqslant\frac{1}{2}\text{ for }i\in N' \biggr\},
\end{equation}
\tag{6.3}
$$
here, $x=\sum_{i\in N'}x_ib_i$ and $N'=N\cup\{n+1\}$. Note that the lateral facets $\pm F(b_i)$ for $i\in N$ are independent of $h$. Hence the DV-cell $P^{n+1}(h)$ is contained in the infinite $(n+1)$-dimensional prism with base $Q^n$ for all $h$. Let $\pm F(S,h)$ be the facet with facet vectors from the second set in ${\mathcal F}(h)$ and of norm $n/4+h^2$. These facets bound the $(n+1)$-dimensional ortahedron $O^{n+1}(h)$ elongated along the vector $e$. In (6.1), we set
$$
\begin{equation*}
x=\sum_{i\in N'}x_ib_i\quad \text{and}\quad l=\frac{1}{2}(b(S)-b(N\setminus S)\pm b_{n+1}).
\end{equation*}
\notag
$$
Setting $x(S)=\sum_{i\in S}x_i$, we have
$$
\begin{equation}
O^{n+1}(h)=\biggl\{x\in\mathbb{R}^n\colon x(S)-x(N\setminus S)\pm 4h^2x_{n+1}\leqslant\frac{n}4+h^2 \text{ for }S\subseteq N\biggr\}.
\end{equation}
\tag{6.4}
$$
It is easily checked that $O^{n+1}(0)=\mu_nO^n$, where $\mu_n=n/2$, and $O^n$ is a regular $n$-dimensional ortahedron dual to the cube. Note that, for all values of the height $h\geqslant 0$, the facet vectors of the DV-cell $P^{n+1}(h)$ are contained in the set ${\mathcal F}(h)$ (see (6.2)). Hence we reach the following result. Next, let us describe how the DV-cell $P^{n+1}(h)$ varies with increasing height $h$. Consider the hyperplane
$$
\begin{equation*}
H_{1/4}=\biggl\{\sum_{i\in N}x_ib_i+x_{n+1}b_{n+1}\in\mathbb{R}^{n+1}\colon \sum_{i\in N}x_ib_i\in\mathbb{R}^n,\, x_{n+1}=\frac14\biggr\}.
\end{equation*}
\notag
$$
It should be noted that since the vector $b_{n+1}=2he$ depends on $h$, the position of the hyperplane $H_{1/4}$ is also a function of $h$. This hyperplane intersects the vertical axis at the point which has only one non-zero coordinate $x_{n+1}=1/4$ in the basis ${\mathcal B}^{n+1}(h)$, that is, at the point $b_{n+1}/4=he/2$. So, the hyperplane $H_{1/4}$ lies precisely between the zero and the first layers of the lattice $L^{n+1}_Z(h)$. Recall that the coordinate of each point of the first layer is $x_{n+1}=1/2$. The vectors of the form $b(S)-b(N)/2+b_{n+1}/2$ are facet vectors of the facet $F(S,h)$ of the upper cap of the ortahedron $O^{n+1}(h)$. The facets $F(S,h)$ change from the vertical position for $h=0$ to the horizontal direction with $h\to\infty$. The facets $F(S,h)$ are inclined so that their intersections with the hyperplane $H_{1/4}$ remain intact. Indeed, using (6.4), it is easily checked that the intersection
$$
\begin{equation*}
O^{n+1}(h)\cap H_{1/4}=\biggl\{x\in{\mathbb R}^n \colon x(S)-x(N\setminus S)\leqslant \frac{n}{4}\ \ \text{for} \ \ S\subseteq N\biggr\}=O^{n+1}(0)=\mu_nO^n
\end{equation*}
\notag
$$
is a regular $n$-dimensional ortahedron independent of $h$. It is clear that $Q^{n+1}(h)\cap H_{1/4}=Q^n$. We have $Q^n\cap\mu_nO^n=P(D_n^*)$ (see the introduction), and so we reach the following result. The DV-cell is centrally symmetric, and so a similar equality also holds for the intersection with the hyperplane $H_{-1/4}$, which is centrally symmetric to the hyperplane $H_{1/4}$. This should be expected because, for $h=0$, the hyperplanes $H_{\pm 1/4}$ coincide with the hyperplane of the zero layer of the lattice $L^{n+1}_Z(h)$, and $P^{n+1}(0)=P(D_n^*)$. If $h>0$, then the support hyperplanes of the facets $F(S,h)$ of the upper cap intersect, for all $S\subseteq N$, at the point $v^0(h)=x^0_{n+1}b_{n+1}$, which lies on the central vertical axis spanned by the vector $e$. For the coordinates of this point, the inequality in (6.4) becomes an equality, and hence
$$
\begin{equation*}
x^0_{n+1}=(2h)^{-2}\biggl(\frac{n}4+h^2\biggr)\quad\text{and}\quad v^0(h)=\frac{n/4+h^2}{(2h)^2}b_{n+1}.
\end{equation*}
\notag
$$
If $h^2=h^2_n=n/4$, then $x^0_{n+1}=1/2$, and, by (6.3), the point $v^0(h_n)=b_{n+1}/2$ lies on the upper facet $F(b_{n+1})$ of the elongated cube $Q^{n+1}(h)$. Hence, for $h^2=n/4$, the vector $b_{n+1}$ from the facet vector becomes a contact vector, and the facet $F(b_{n+1})$ becomes a contact face (a contact vertex, to be more precise). For $h^2>n/4$, the vector $b_{n+1}$ also ceases to be a contact vector. So, for the critical value $h^2=h_n^2=n/4$, there occurs the above modification of the DV-cell $P^{n+1}(h)$. Similarly, for other critical values of $h=h_s$, where $m\leqslant s\leqslant n-1$, a modification of the DV-cell $P^{n+1}(h)$ occurs, for which some faces corresponding to contact vectors of the compound class $C(S)$, where $|S|=s$, change their status of a contact face. The value $h=1/2$ plays a special role. Indeed, for $h=1/2$, each contact vector in Table 1 has either all integer coordinates or all half-integer coordinates in the basis ${\mathcal B}^{n+1}(1/2)$. The vectors in Table 1 generate the entire lattice, and hence this result holds for all vectors of the lattice $L^{n+1}(1/2)$, which is nothing else as the lattice $D^*_{n+1}$ (see the introduction). In addition, we have
$$
\begin{equation*}
Q^{n+1}\biggl(\frac12\biggr)=Q^{n+1}, \qquad O^{n+1}\biggl(\frac12\biggr)=\mu_{n+1}O^{n+1},
\end{equation*}
\notag
$$
where $Q^{n+1}$ and $\mu_{n+1}O^{n+1}$ are the $(n+1)$-dimensional cube and ortahedron, respectively. Now by Proposition 4 we have $P^{n+1}(1/2)=P(D^*_{n+1})$. Recall that, for $h=0$, the lattice $L^{n+1}(0)$ has dimension $n$ and is a centering of the original lattice $Z^n$. Since $Z^n$ is centred by the fractional vectors $v(S)=b(S)-\frac{1}{2}b(N)$ for all $S\subseteq N$, this centering is $D_n^*$. Thus, we have reached the following result. Proposition 5. $L^{n+1}(0)=D_n^*$ and $L^{n+1}(1/2)=D_{n+1}^*$. Hence $P^{n+1}(0)=P(D_n^*)$ and $P^{n+1}(1/2)=P(D_{n+1}^*)$. § 7. The DV-cell $P^{n+1}(h)$ for $h^2\geqslant n/4$Let us show that, for $h^2\geqslant n/4$, the DV-cells $P^{n+1}(h)$ are zonotopes. Recall that the zonotope $Z_{\beta}(U)$ is the weighted Minkowskii sum with weights $\beta_u$ of the vectors of a set $U$. To be more precise,
$$
\begin{equation*}
Z_{\beta}(U)=\biggl\{x\in\mathbb{R}^n\colon x=\sum_{u\in U}\lambda_uu,\, -\frac{1}{2}\beta_u\leqslant \lambda_u\leqslant\frac{1}{2}\beta_u\biggr\}.
\end{equation*}
\notag
$$
This zonotope is a parallelohedron if and only if $U$ is a unimodular set of vectors. We let $Z(U)$ denote the zonotope $Z_{\beta}(U)$ with $\beta_u=1$ for all $u\in U$. Recall that a set of vectors $U\subset{\mathbb R}^n$ is unimodular if each vector from $U$ can be expressed as an integer combination of vectors of any of its basis subset. Two unimodular sets $U$ and $U'$ are called equivalent if $U'=QU$, where $Q$ is a non-degenerate matrix. In the present paper, we deal only with unimodular sets which are equivalent to inclusion-maximal subsets (in its dimension) of the well-known set of roots ${\mathbb A}_n$. This is a graphical unimodular set corresponding to the complete graph $K_{n+1}$ on $n+1$ vertices. This means that one can choose signs of the roots of the set ${\mathbb A}_n$ so that with these roots one can associate edges of the graph $K_{n+1}$ so that the sum of the roots associated with edges of an arbitrary cycle of the graph is zero. Hence to each $U\subseteq{\mathbb A}_n$ there corresponds the graph $G(U)\subseteq K_n$. For more details, see, for example, [9]. The zonotope $Z({\mathbb A}_n)$ is a primitive DV-cell. In [10] and [11], it was shown that the parallelohedron $Z_{\beta}(U)$ is described by linear inequalities as follows:
$$
\begin{equation}
Z_{\beta}(U)=\biggl\{x\in\mathbb{R}^n\colon \langle p,x\rangle\leqslant\frac{1}{2}\,\varphi_U(p)\text{ for all }p\in U^*\biggr\},
\end{equation}
\tag{7.1}
$$
where $\varphi_U(p)=\sum_{u\in U}\beta_u\langle p,u\rangle^2$ and
$$
\begin{equation}
U^*=\bigl\{p\in\mathbb{R}^n\colon \langle p,u\rangle\in\{0,\pm 1\}\text{ for all }u\in U\bigr\}
\end{equation}
\tag{7.2}
$$
is the set dual to $U$, and, in addition, $(U^*)^*=U$. According to the previous section, for $h\geqslant h_n$, where $h_n^2=n/4$, the DV-cell $P^{n+1}(h)$ has the following set of facets
$$
\begin{equation*}
\{\pm F(S,h)\colon S\subseteq N\}\cup\{\pm F(b_i)\colon i\in N\},
\end{equation*}
\notag
$$
where the first set consists of facets of the upper (+) and lower $(-)$ caps, and the second set contains the lateral facets, which form the equator of the DV-cell $P^{n+1}(h)$. The set of contact vectors of the DV-cell $P^{n+1}(h)$, which are different from the facet vectors, is the set of vectors of the first row of Table 1, and only for $h=h_n$ it contains the vectors $\pm b_{n+1}$. Hence
$$
\begin{equation}
\mathcal{K}(h)=\mathcal{F}(h)\cup\{b(S)-2b(T)\colon T\subseteq S \subseteq N\},
\end{equation}
\tag{7.3}
$$
where the set ${\mathcal F}(h)$ (see (6.2)) is the set of all contact vectors of the DV-cell $P^{n+1}(h)$ for $h\geqslant h_n$. Let us show that, for $h\geqslant h_n$, the DV-cell $P^{n+1}(h)$ is a zonotope. To this end, we find the dual set
$$
\begin{equation}
\mathcal{K}(h)^*=\bigl\{u\in\mathbb{R}^{n+1}\colon \langle p,u\rangle\in\{0,\pm 1\}\text{ for all }p\in\mathcal{K}(h)\bigr\}.
\end{equation}
\tag{7.4}
$$
Let us use the basis ${\mathcal B}^{n+1}(h)=\{b_i \colon i\in N'\}$, where $N'=N\cup\{n+1\}$ and $b_{n+1}=2he$. By going through all coordinates $u_i$ of the vector $u=\sum_{i\in N'}u_ib_i$, one can easily show that, for $h=h_n$, the vectors $u$ that satisfy (7.4) compose the set
$$
\begin{equation*}
\mathcal{K}(h)^*=U_0^{n+1}(h)=\biggl\{\frac{b_{n+1}}{(2h)^2}\pm b_i\colon i\in N\biggr\}.
\end{equation*}
\notag
$$
This set is a unimodular set of vectors of graphical type. The corresponding graph $G(U_0^{n+1})=G(n)$ consists of $n$ two-edge chains corresponding to the vectors $b_{n+1}/(2h)^2+b_i$ and $b_{n+1}/(2h)^2- b_i$ with two common terminal vertices and $n$ different interior vertices. For $h>h_n$, the vectors $\pm b_{n+1}$ are removed from the set ${\mathcal K}(h)$ of contact vectors, and the unimodular set $U_0^{n+1}(h)$ is augmented with the vector $b_{n+1}/(2h^2)$. So, for $h>h_n$,
$$
\begin{equation*}
\mathcal{K}(h)^*=U_h^{n+1}=U_0^{n+1}(h)\cup\biggl\{\frac{e}{h}\biggr\}.
\end{equation*}
\notag
$$
This unimodular set is also of graphical type, and $G(U_h^{n+1})=G(n,e)$. In the graph $G(n,e)$, all two-edge chains of the set $U_0^{n+1}(h)$ are closed by a new edge corresponding to the new vector $e/h$, which is equal to the sum of the vectors representing the edges of the two-edge chain. In view of the result obtained at the beginning of this section, we have the following assertion. Proposition 6. Let $P^{n+1}(h)$ be a DV-cell of the lattice $L^{n+1}(h)$. Then $P^{n+1}(h_n)$ is the zonotope $Z_{\beta}(U_0^{n+1}(h_n))$, where $h_n^2=n/4$, and $P^{n+1}(h)$ is the zonotope $Z_{\beta}(U_h^{n+1})$ if $h>h_n$. Since $Z_{\beta}=P^{n+1}(h)$, where $h^2\geqslant n/4$, is a classical DV-cell, the zonotope $Z_{\beta}$ also has the description (6.1). Hence the right-hand side of description (7.1) should involve the function $p^2/2$, that is, the equality $\varphi_U(p)=p^2$ should be satisfied. This condition is satisfied by the form
$$
\begin{equation*}
\varphi_U(p)=\frac{1}{2}\sum_{u\in U_0^{n+1}(h)}\langle p,u\rangle^2+ \biggl(h^2-\frac{n}{4}\biggr)\biggl\langle p,\frac{e}{h}\biggr\rangle^2.
\end{equation*}
\notag
$$
Indeed, for $h^2\geqslant n/4$, the DV-cell $P^{n+1}(h)$ has two types of normal vectors, namely, the vectors $p(S)=b(S)-b(N)/2\pm he$ of norm $p(S)^2=n/4+h^2$ for all $S\subseteq N$ and the basis vectors $\pm b_i$ of norm $b_i^2=1$ for all $i\in N$. In addition, it has the contact vectors $b(S)-2b(T)$, where $T\subseteq S$, of norm $|S|$ for all $S\subseteq N$, the vectors $\pm b_{n+1}$ of norm $4h^2$, and the vectors of the parity class $b(N)-2b(T)$, which is a contact vector only if $h^2=n/4$. A direct verification shows that $\varphi_U(p)=p^2$ for all $p\in U^*$ if $\beta_u=1/2$ for $u\in U^{n+1}_0(h)$ and $\beta_u=h^2-n/4$ for $u=e/h$. Recall that, for $h^2>n/4$, the DV-cell $P^{n+1}(h)$ is the sum of the vectors $(e/(2h)\pm b_i)/2$ for all $i\in N$ and of the vector $(h^2-n/4)e/h$. If $h\to\infty$, then the first vectors tend to $\pm b_i/2$ for $i\in N$, and the second vector tends to the vector $he$, whose length tends to infinity. Hence the DV-cell itself tends, as $h\to\infty$, to the $(n+1)$-dimensional prism of height $h$ and with cube $Q^n$ in the base. At the end of the previous section it was noted that, for $h\geqslant h_n$, the DV-cell $P^{n+1}(h)$ lies inside the cube $Q^{n+1}(h)$. In addition to the facets $\pm F(S,h)$ of the upper and lower caps, $P^{n+1}(h)$ have only lateral facets $\pm F(b_i)$, which lie in the cube $Q^{n+1}(h)$. Hence the projection along the vector $e$ splits the original cube $Q^n$ into $2^n$ small cubes, which are the projections of $2^n$ facets $F(S,h)$ for all $S\subseteq N$. The next result verifies this claim using the graphical structure of the unimodular sets $U^{n+1}_0(h)$ and $U^{n+1}_h$. Proposition 7. Let $h\geqslant h_n$. Then the facets $F(S,h)$ of the upper cap of the DV-cell $P^{n+1}(h)$ are $n$-dimensional parallelepipeds for all $S\subseteq N$, and the lateral facets $\pm F(b_i)$ for all $i\in N$ are affinely equivalent to the $n$-dimensional DV-cell $P^n(h)$. Proof. The facets of the DV-cell $P^{n+1}(h)$ are also zonotopes. They are defined by subsets of the unimodular sets $U_0^{n+1}(h)$ and $U_h^{n+1}$. These subsets consist of the vectors $u$ corresponding to the edges of the graph obtained from the original graph by removal of all edges of the minimal cut. This cut consists of the edges such that, after removal of these edges, the graph splits into two connected components.
Recall that each of the graphs $G(n)$ and $G(n,e)$ consists of $n+2$ vertices, of which $n$ vertices are of degree 2, and the remaining 2 vertices are of degree $n$ and $n+1$, respectively. In our case, there are only two types of cuts — these being single-vertex cuts consisting of the edges incident to vertices of two types. After removal of a single-vertex cut, there remains a graph whose one component consists of a single vertex. Note that our graphs are symmetric with respect to permutations of edges in each two-edge chain. Hence, in each cut corresponding to vertices of degree $n$ or $n+1$, there may appear any edge from the two-edge chain. The facets of the form $F(S,h)$ correspond to single-vertex cuts of vertices of degree $n$ and $n+1$. In both cases, the graph that remains after the removal of the cut is a tree consisting of $n$ edges. The set of vectors corresponding to edges of the tree is linearly independent. The corresponding $n$ vectors $u$ have the form $e/(2h)\pm b_i$, where the plus sign corresponds to $i\in S$, and the minus sign corresponds to $i\in N\setminus S$. It follows that the zonotope corresponding to the facet $F(S,h)$ is a parallelepiped. The facets of the form $F(b_i)$ correspond to single-vertex cuts of vertices of degree 2. The graph which remains after the removal of the cut is either $G(n-1)$ or $G(n-1,e)$. These graphs correspond to unimodular sets whose dimension is one less than that for the sets defining the DV-cell $P^n(h)$. This proves the claims. $\Box$ § 8. The DV-cells $P^{n+1}(h)$ for $n=3$Consider the original dimension $n=3$, that is, consider the lattice $L^4_Z(h)$. In this case, the DV-cells $P^4(h)$ lies in the set of known parallelohedra in the dimension 4. These parallelohedra are either zonotopes $Z(U)$ or the Minkowski sum of the DV-cell $P(D_4)$ with a zonotope. According to Proposition 5, $P^4(0)=P(D_3^*)$ and $P^4(1/2)=P(D_4^*)=P(D_4)$. By Proposition 1 with $n=3$, the number of intervals of variation of $h$ is 6. Recall that $m=\lfloor 3/2\rfloor=2$. In addition, we have $h^2_s=(2s-n)/4$, where the integer $s$ varies in the range $m\leqslant s\leqslant n$, that is, $s=2,3$. Hence, we have two critical values $h^2_2=1/4$ and $h^2_3=3/4$. There are six intervals of variation of $h$:
$$
\begin{equation*}
h=0,\quad 0<h<\frac12,\quad h=\frac12,\quad \frac12<h<h_3,\quad h=h_3,\quad h_3<h,\ \text{ where } \ h^2_3=\frac34.
\end{equation*}
\notag
$$
Here, there is only one mainstay parallelohedron $P^4(1/2)=P(D_4^*)$ for $h=h_2=1/2$, and there is only one value $s=2$ for the cardinality of the set $S\subset N$ for which the parity class $C(S)$ is compound. We set $N=N_3=\{ijk\}$. According to (4.3), the sets of contact vectors of the compound class $C(\{ij\})$ splits into two subsets
$$
\begin{equation*}
\mathcal{L}_0(\{ij\})=\{\pm b_i\pm b_j\}\quad\text{and}\quad \mathcal{L}_h(\{ij\})=\{\pm b_k\pm b_4\}.
\end{equation*}
\notag
$$
In our case, the remarkable property of this decomposition is that
$$
\begin{equation*}
\bigcup_{\{ij\}\subset N_3}(\pm b_i\pm b_j)=\mathbb A_3\quad\text{and}\quad \bigcup_{k\in N_3}(\pm b_k\pm b_4)=U^4_0
\end{equation*}
\notag
$$
are the unimodular sets $U$ such that the DV-cells $P^4(h)$ have the form of the sum $P(D_4^*)+Z(U)$. It is known that each 4-dimensional parallelohedron is the Minkowskii sum of the DV-cell $P(D_4)=P(D_4^*)$ and of the zonotope $Z(U)$. (The paper [12] lists all the unimodular sets $U$ such that the sum $P(D_4)+Z(U)$ is parallelohedron.) There exist 52 4-dimensional parallelohedra. B. N. Delaunay assigned the number $N_D$ to each of these parallelohedra (except one). These numbers can be found in [4]. Below, the DV-cell $P^4(h)$ (except the first one) of each interval of variation of $h$ is marked with the Delaunay number. With the use of Table 2, which is given at the end of [12], it proved possible to find all 6 types of DV-cells $P^4(h)$. In this table, the two graphs $G(3)$ and $G(3,e)$ (from the previous section are denoted, following J. H. Conway, by $C_{222}$ and $C_{222}+{\bf 1}$, respectively (see [13], where the 4-dimensional lattices are considered at the end of the third lecture). The corresponding DV-cells are as follows: (1) $P^4(0)=Z({\mathbb A}_3)=Z(D_3^*)=Q^3\cap\mu_3O^3$ is the 3-dimensional primitive truncated octahedron. Recall that $P(D_n^*)$ is the truncated ortahedron; (2) $P^4(h)=\alpha(h)Z({\mathbb A}_3)+\beta(h)P(D_4^*)$ for $0<h<1/2$ has the Delaunay number $N_D=33$; (3) $P^4(1/2)=P(D_4^*)\simeq P(D_4)$ with Delaunay number $N_D=51$. This is a unique mainstay parallelohedron in the dimension 4; (4) $P^4(h)=\beta(h)P(D_4^*)+\gamma(h)Z(U^4_0)$ for $1/4<h^2<3/4$ has the Delaunay number $N_D=31$; (5) $P^4(h)=Z(U^4_0)$ for $h^2=3/4$ has the Delaunay number $N_D=11$; (6) $P^4(h)=Z(U^4_h)$ for $h^2>3/4$ has the Delaunay number $N_D=9$. Here, the functions $\alpha(h),\beta(h),\gamma(h)$ assume the following values at the end-points of the corresponding intervals:
$$
\begin{equation*}
\begin{gathered} \, \alpha(0)=1, \quad \alpha\biggl(\frac12\biggr)=0, \quad\beta(0)=0,\quad \beta\biggl(\frac12\biggr)=1, \quad \beta(h_3)=0, \\ \gamma\biggl(\frac12\biggr)=0, \quad\gamma(h_3)=1. \end{gathered}
\end{equation*}
\notag
$$
It is worth noting that in [4] the sketches with numbers $N_D=11$ and $N_D=9$ differ from that into 8 cubes, which was mentioned in the introduction. This is because Delaunay projects the lifted DV-cell $P^{n+1}(h)$ along the vector $u\in U^4_{0,h}$, which is different from $e$. 
Citation in format AMSBIB
\Bibitem{Gri24} |
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