MATHEMATICS

E. P. BARANOVSKII

LOCAL MINIMA OF THE DENSITY OF A LATTICE COVERING OF FOUR-DIMENSIONAL EUCLIDEAN SPACE BY EQUAL SPHERES

(Presented by Academician S. L. Sobolev on 10 V 1965)

Let a point lattice (\Gamma) be given in four-dimensional Euclidean space (E^4). As parameters of the lattice we choose the scalar products (g_{kl}) ((k,l=1,2,\ldots,5;\ k\ne l)) of the vectors (\mathbf a_k) of a Selling frame (\left({}^{1}\right)) of this lattice. Denote by (D_4(\Gamma)) the density of the covering of the space (E^4) by spheres of equal radii with centers at the points of the lattice (\Gamma).

The present note is devoted to the problem of finding the local minima of the density (D_4(\Gamma)) as a function of the parameters (g_{kl}) of the lattice (\Gamma), and the problem considered is solved completely. Known results directly or indirectly related to the question of local minima of (D_4(\Gamma)) are contained in papers (\left({}^{2-6}\right)).

1. Consider the star (\gamma) of the simplicial (L)-decomposition (\left({}^{1}\right)) of the lattice (\Gamma), i.e., the set of (L)-simplices having as a common vertex an arbitrarily chosen point (O) of the lattice (\Gamma). A hyperplane (P=(O,\mathbf a_k,\mathbf a_l,\mathbf a_m)) passing through the center (O) of the star (\gamma) and the vectors (\mathbf a_k,\mathbf a_l,\mathbf a_m) of a Selling frame will be called a hyperplane of symmetrization of the star (\gamma) (and of the lattice (\Gamma)) if, with respect to it, the star (\gamma) is affinely symmetric, i.e., there exists an affine transformation of the space (E^4) preserving the hyperplane (P), which carries the star (\gamma) into a star (\tilde{\gamma}) symmetric with respect to the hyperplane (P). Everywhere here Selling frames are assumed to be reduced in the sense of Voronoi (\left({}^{7,8}\right)).

Lattices on which local minima of the density (D_4(\Gamma)) are attained will be called extremal lattices. A necessary condition that a lattice (\Gamma) must satisfy in order to be extremal follows from the following theorem:

Theorem 1. If the star (\gamma) of a four-dimensional lattice (\Gamma) is not symmetric with respect to any of its hyperplanes of symmetrization, then the lattice (\Gamma) is not extremal.

We indicate the course of the proof of Theorem 1. The density (D_4(\Gamma)) is equal to

[
D_4(\Gamma)=\frac{\pi^2}{2}\left(\max_{\nu=1,2,\ldots,12} R_\nu\right)^4 V^{-1},
\tag{1}
]

where (R_\nu) ((\nu=1,2,\ldots,12)) denotes the radii of the spheres circumscribed about each of the 12 nonhomologous (L)-simplices of the lattice (\Gamma), and (V) denotes the volume of the fundamental parallelepiped of the lattice (\Gamma). Consider the set of shear transformations preserving the hyperplane of symmetrization (P) and the volume (V). We assume that each of the transformations considered changes the parameters (g_{kl}) of the lattice (\Gamma) arbitrarily little.

If the star (\gamma) is not symmetric with respect to (P), then among the indicated shear transformations there is always one such that (\max_{\nu=1,2,\ldots,12} R_\nu) for the varied lattice (\Gamma^*) will be smaller than the same quantity for the initial lattice (\Gamma). After this, the assertion of the theorem follows from formula (1).

1. In the space (G^{10}) of the parameters (g_{kl}), the reduction domain (G_{\mathrm{I}}) of lattices of the first type, in which we shall consider the function (D_4(\Gamma)), is defined by the inequalities ((^1,^8))

[
g_{kl} \leqslant 0,\qquad (k,l=1,2,\ldots,5;\ k\ne l).
\tag{2}
]

The star (\gamma) of lattices of the first type is constructed in such a way that any of the hyperplanes (P=(O,\mathbf a_k,\mathbf a_l,\mathbf a_m)) ((k,l,m=1,2,\ldots,5;\ k\ne l,\ l\ne m,\ m\ne k)) is a hyperplane of symmetrization. A necessary and sufficient condition for the symmetry of the star (\gamma) with respect to the hyperplane ((O,\mathbf a_{k_1},\mathbf a_{k_2},\mathbf a_{k_3})) is the fulfillment, for the lattice parameters, of the three equalities:

[
g_{k_i k_4}=g_{k_i k_5}\qquad (i=1,2,3).
\tag{3}
]

From Theorem 1 and relations (3) it follows that the only lattice that can be extremal on the set of lattices of the first type is a unique lattice (we consider similar lattices to be identical), namely the so-called principal lattice (\Gamma_1^4) of the first type, for which all parameters (g_{kl}) are equal to one another. The extremality of the lattice (\Gamma_1^4) was proved by Gametskii ((^2,^3)).

Thus, on the set of lattices of the first type the question of local minima of (D_4(\Gamma)) is completely solved:

Theorem 2. In the domain (G_{\mathrm{I}}) of lattices of the first type there exists a unique local minimum of the density (D_4(\Gamma)), attained on the lattice (\Gamma_1^4), and

[
D_4(\Gamma_1^4)=\frac{2\pi^2}{\sqrt{125}}=1.7655\ldots .
\tag{4}
]

1. One of the reduced domains (G_{\mathrm{II}}) of lattices of the second type is specified by the inequalities (((^8), p. 363))

[
g_{45}\geqslant 0,\qquad
g_{ij}\leqslant 0,\qquad
g_{45}+g_{i4}\leqslant 0,\qquad
g_{45}+g_{i5}\leqslant 0
\qquad (i,j=1,2,3;\ i\ne j).
\tag{5}
]

The star (\gamma) of lattices of this type has already only 4 hyperplanes of symmetrization: (P_i=(O,\mathbf a_i,\mathbf a_4,\mathbf a_5)) ((i=1,2,3)) and (P_4=(O,\mathbf a_1,\mathbf a_2,\mathbf a_3)).

On the basis of Theorem 1 and equalities (3) we conclude that the extremal lattices of the second type can only be those to which, in the space (G^{10}), there correspond points lying at the intersection of the domain (G_{\mathrm{II}}) with the three-dimensional plane:

[
g_{14}=g_{15}=g_{24}=g_{25}=g_{34}=g_{35}=g,\qquad
g_{12}=g_{13}=g_{23}=h,\qquad
g_{45}=t.
\tag{6}
]

The intersection of the domain (G_{\mathrm{II}}) with the plane (6) has been investigated ((^5)). Combining the result obtained there with ours, we obtain the theorem:

Theorem 3. In the domain (G_{\mathrm{II}}) of lattices of the second type, the density (D_4(\Gamma)) has a unique local minimum, which is attained on the lattice (\Gamma_2^4) with parameters (g_{kl}) satisfying the conditions (6) and such that

[
g:h:t=-1:-0.5441\ldots:0.5001\ldots,
\tag{7}
]

where the numbers (x=-h/g) and (y=-t/g) constitute one of the solutions of the system

[
xy^2-2x^2-3xy+2(x+y)=0,
]

[
x(9y^2-24y+7)-14y^2+40y-18=0.
\tag{8}
]

On the lattice (\Gamma_2^4) the density is equal to:

[
D_4(\Gamma_2^4)=\frac{\pi^2}{2}\cdot 0.3817\ldots=1.8836\ldots .
\tag{9}
]

1. Let us consider the set of lattices of the third type. In the space (G^{10}), one of the reduced domains (G_{\mathrm{III}}) of lattices of the third type is defined by means of the inequalities ((8), p. 366)

[
\begin{gathered}
g_{45} \geqslant 0,\qquad g_{14} \leqslant 0,\qquad g_{15} \leqslant 0,\qquad g_{23}-g_{45}\geqslant 0,\
g_{12}+g_{23}-g_{45}\leqslant 0,\qquad
g_{13}+g_{23}-g_{45}\leqslant 0,\qquad
g_{24}+g_{23}\leqslant 0,\
g_{25}+g_{23}\leqslant 0,\qquad
g_{34}+g_{23}\leqslant 0,\qquad
g_{35}+g_{23}\leqslant 0 .
\end{gathered}
\tag{10}
]

For the star (\gamma) of lattices of the third type there are two hyperplanes of symmetrization:
(P'=(O,a_1,a_2,a_3)) and (P''=(O,a_1,a_4,a_5)). From Theorem 1 and equalities (3) we obtain that the extremal lattices may be those which correspond to points situated at the intersection of the (\widetilde{G}{\mathrm{III}}) domain (G) with the five-dimensional plane}

[
g_{24}=g_{25}=g_{34}=g_{35}=g,\quad
g_{12}=g_{13}=h_1,\quad
g_{14}=g_{15}=h_2,\quad
g_{23}=t,\quad
g_{45}=u .
\tag{11}
]

Lattices whose parameters satisfy (11) have, in general, 6 different radii (R_\nu) ((\nu=1,2,\ldots,12)).

The usual investigation of the function (1) in the domain (\widetilde{G}_{\mathrm{III}}) gives the following result:

Theorem 4. On the set of lattices of the third type there exists a unique extremal lattice (\Gamma_3^4), whose parameters (g_{kl}) are determined by the relations

[
g_{12}=g_{13}=-g_{23}=\alpha,\qquad
g_{14}=g_{15}=-g_{45}=\beta,
\tag{12}
]

[
g_{24}=g_{25}=g_{34}=g_{35}=\alpha+\beta,\qquad
\alpha:\beta=(1+\sqrt{13}):2,\qquad
\alpha<0,\quad \beta<0,
]

with

[
D_4(\Gamma_3^4)=\frac{\pi^2}{2}\cdot 0.39085\ldots=1.9287\ldots .
\tag{13}
]

Theorems 2–4 give a complete answer to the question of local minima of density.

The author takes the opportunity to express deep gratitude to Corresponding Member of the Academy of Sciences of the USSR B. N. Delone and to S. S. Ryshkov for discussion of the results obtained and for repeated assistance in carrying out this work, and to S. V. Smirnov for constant attention.

Received
15 IV 1965

CITED LITERATURE

1. B. N. Delone, UMN, vol. 3, 16 (1937).
2. A. F. Gametskii, DAN, 146, No. 5 (1962).
3. A. F. Gametskii, DAN, 151, No. 3 (1963).
4. B. N. Delone, S. S. Ryshkov, DAN, 152, No. 3 (1963).
5. E. P. Baranovskii, S. S. Ryshkov, Abstracts of Reports of the Second All-Union Geometric Conference, Kharkov, 1964.
6. S. S. Ryshkov, DAN, 162, No. 2 (1965).
7. B. N. Delone, UMN, vol. 4, 102 (1938).
8. G. F. Voronoi, Collected Works, 2, Kiev, 1952, pp. 239–368.