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**The kissing number in four dimensions.**
*(English)*
Zbl 1169.52008

For each \(n \geq 1\), the kissing number \(k(n)\) is defined to be the largest number of equal non-overlapping spheres in \(\mathbb R^n\) that can touch another sphere of the same size. The value of \(k(n)\) has been known for \(n=1,2,3\) and (somewhat surprisingly, due to unique features of \(E_8\) and the Leech lattice) for \(n=8,24\):
\[
k(1)=2,\;k(2)=6,\;k(3)=12,\;k(8)=240,\;k(24)=196560.
\]
In fact, the value \(k(3)\) was the subject of a famous discussion between Isaac Newton and David Gregory, and was known under the name of the problem of 13 spheres until its resolution by Schütte and van der Waerden in 1953.

In four dimensions, it has long been known that the kissing number of the lattice \(D_4\) is 24, while the upper bound obtained by A. M. Odlyzko and N. J. A. Sloane [J. Comb. Theory, Ser. A 26, 210–214 (1979; Zbl 0408.52007)] by an application of Delsarte’s method guaranteed that \(k(4) \leq 25\). In the paper under review, the author presents a powerful extension of Delsarte’s linear programming method, obtaining a bound \(k(24) < 25\). This shows that \(k(4)=24\), solving a long-standing open problem.

It is well-known that the kissing number problem in \(\mathbb R^n\) can be reformulated to ask how many points can be placed on the surface of the sphere \(\mathbb S^{n-1}\) so that the angular separation between any two of these points is at least \(\pi/3\). To approach this question, the author uses the linear programming method, obtaining a certain polynomial \(f_4(t)\) of degree 9, which has the following remarkable property: for each set of \(M\) points \(\{ x_1,\dots,x_M \}\) on the sphere \(\mathbb S^3\) such that the angular separation between any two of these points is \(\pi/3\), \[ M^2 \leq \sum_{i=1}^M \sum_{j=1}^M f_4(x_i \cdot x_j) < 25 M. \] Clearly this inequality cannot be satisfied for any \(M \geq 25\), which completes the proof.

This important paper is very well written and provides a nice survey of the area.

In four dimensions, it has long been known that the kissing number of the lattice \(D_4\) is 24, while the upper bound obtained by A. M. Odlyzko and N. J. A. Sloane [J. Comb. Theory, Ser. A 26, 210–214 (1979; Zbl 0408.52007)] by an application of Delsarte’s method guaranteed that \(k(4) \leq 25\). In the paper under review, the author presents a powerful extension of Delsarte’s linear programming method, obtaining a bound \(k(24) < 25\). This shows that \(k(4)=24\), solving a long-standing open problem.

It is well-known that the kissing number problem in \(\mathbb R^n\) can be reformulated to ask how many points can be placed on the surface of the sphere \(\mathbb S^{n-1}\) so that the angular separation between any two of these points is at least \(\pi/3\). To approach this question, the author uses the linear programming method, obtaining a certain polynomial \(f_4(t)\) of degree 9, which has the following remarkable property: for each set of \(M\) points \(\{ x_1,\dots,x_M \}\) on the sphere \(\mathbb S^3\) such that the angular separation between any two of these points is \(\pi/3\), \[ M^2 \leq \sum_{i=1}^M \sum_{j=1}^M f_4(x_i \cdot x_j) < 25 M. \] Clearly this inequality cannot be satisfied for any \(M \geq 25\), which completes the proof.

This important paper is very well written and provides a nice survey of the area.

Reviewer: Lenny Fukshansky (Claremont)

### MSC:

52C17 | Packing and covering in \(n\) dimensions (aspects of discrete geometry) |

52B40 | Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.) |

52C07 | Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry) |

52C15 | Packing and covering in \(2\) dimensions (aspects of discrete geometry) |

52C45 | Combinatorial complexity of geometric structures |