MATHEMATICS

Corresponding Member of the USSR Academy of Sciences B. N. DELONE, S. S. RYSHKOV

SOLUTION OF THE PROBLEM OF THE LEAST DENSE LATTICE COVERING OF FOUR-DIMENSIONAL SPACE BY EQUAL SPHERES

The problem considered here (trivial in the case of the plane) already presents difficulties for three-dimensional space, for which it was first solved by the Indian mathematician Bambah ((^1)). For four-dimensional space this problem had not had a solution up to the present time. For a bibliography of this and related questions, see ((^2,\,^3)).

1. Using Voronoi’s tables ((^4)) for the structure of stars ((L)) and Dirichlet regions of all three types of four-dimensional lattices, one can obtain the following expression for the sum of the squares of the radii of the Dirichlet regions of a lattice:

[
\sum_{\lambda=1}^{120} R_\lambda^2=\frac{24}{5}A+2C.
]

Here (C) denotes the sum of the squares of the distances from the centers of the circumscribed spheres to the centers of gravity of the nonhomologous simplexes (L) of the lattice under consideration. The expression (A) for the first type has the form (A_{\mathrm{I}}=-5\sum_{i<k}g_{ik}); for the second,

[
A_{\mathrm{II}}=-5\sum_{i<k}g_{ik}-3g_{12};
]

for the third,

[
A_{\mathrm{III}}=-5\sum_{i<k}g_{ik}-3g_{12}-2g_{34};
]

here (g_{ik}) are the scalar products of the vectors of the Selling-reduced frame of the lattice under consideration (see the article by B. N. Delone ((^5))).

1. In the 10-dimensional space of the parameters (g_{ik}), the surface (\Delta=\mathrm{const}), where (\Delta) is the determinant of the lattice, has at each of its points a tangent plane and is convex toward the origin of coordinates. Therefore, by virtue of the symmetry of this surface, and also of the plane (A_{\mathrm{II}}=\mathrm{const}) with respect to the three-dimensional plane

[
g_{13}=g_{14}=g_{15}=g_{23}=g_{24}=g_{25},\quad
g_{34}=g_{35}=g_{45},
]

the unique minimum of the function (A_{\mathrm{II}}) for (\Delta=\mathrm{const}) lies in this three-dimensional plane, i.e. the problem reduces to finding the minimum of the linear function (-30g-15h-8x) under

[
g(2x+3g)(2g+3h)^2=\mathrm{const}
]

(for (\Delta=\mathrm{const})). This minimum lies in the region of the second type, and its value (m_2) is greater than the value (m_1) of the minimum of the function (A_{\mathrm{I}}) for determinant (\Delta) equal to the same constant. But since at the point of the minimum of (A_{\mathrm{I}}) the quantity (C) is equal to zero, and in general it is greater than or equal to zero, the sum

[
\sum_{\lambda=1}^{120} R_\lambda^2
]

for any lattice of the second type is greater than for the lattice (\Gamma_1^4) (see ((^2))), in which (A_{\mathrm{I}}) is minimal. But for the lattice (\Gamma_1^4) all radii (R_\lambda) are equal; consequently, the largest radius (R_\lambda) of any lattice of the second type is greater than the radius of the lattice (\Gamma_1^4), i.e. the density of the covering corresponding to any lattice of the second type is greater than the density corresponding to the lattice (\Gamma_1^4).

1. In the case of the third type, we similarly use the symmetry of the surface (\Delta=\mathrm{const}) and of the plane (A_{\mathrm{III}}=\mathrm{const}) with respect to the five-dimensional plane

[
g_{13}=g_{14}=g_{15}=g_{23}=g_{24}=g_{25},\quad
g_{35}=g_{45},
]

and arrive at finding the minimum of the function

[
-20g-10h-10k-8x-7y
]

under

[
(2x+2g+k)(2y+2g+h)(2gh+2gk+kh)=\mathrm{const},
]

i.e. for (\Delta=\mathrm{const}). This minimum

lies in the region of the third type, and its value (m_3) is also greater than (m_1). This, evidently, completes the proof that the lattice (\Gamma_1^4) corresponds to the least dense lattice covering of four-dimensional Euclidean space by equal spheres.

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
13 VIII 1963

REFERENCES

1. R. P. Bambah, Proc. Nat. Inst. Sci. India, 20, 25 (1954).
2. А. Ф. Гамецкий, DAN, 146, No. 5 (1962).
3. А. Ф. Гамецкий, DAN, 151, No. 3 (1963).
4. Г. Ф. Вороной, Collected Works, 2, Kiev, 1952, pp. 341—368.
5. Б. Н. Делоне, UMN, vol. 4, 102 (1938).