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**A proof of the Kepler conjecture.**
*(English)*
Zbl 1096.52010

The paper contains an abridged form of the proof of the famous Kepler conjecture, saying that no packing of congruent balls in Euclidean 3-space has density larger than that of the face-centered cubic packing; this density equals \(\pi/\sqrt{18}\approx 0,74\). The author cites also his other related papers containing the full proof (this one will appear in Discrete and computational Geometry), an expository account of the proof, a discussion of the Computer algorithms used in his approach, or speculations about the structure of a second-generation proof. The important contribution of Samuel P. Ferguson to difficult proof parts is underlined, such that Ferguson is even the coauthor of the key chapter 5 of the paper. The main steps of the proof can be described (very shortly) as follows.

Let \(\Lambda\) denote the set of ball centers in a so-called saturated packing. Denoting by \(B(x,r)\) the closed ball with center \(x\) and radius \(r\), the author introduces \(\sigma(x, r,\Lambda)\) as finite density being the ratio of the volume of \(B(x,r,\Lambda)\) to the volume of \(B(x,r)\) (here \(B(x,r,\Lambda)\) denotes the intersection with \(B(x,r)\) of the union of all balls in the packing). Using Voronoi cells of \(\Lambda\) and certain properties of suitable types of functions, it is shown that there is a constant \(C\) such that for all \(r\geq 1\) and all \(x\in\mathbb R^3\) the inequality \(\sigma(x,r,\Lambda)\leq\pi/\sqrt{18}+C/r\) holds.

Based on this, the precise meaning of Kepler’s conjecture is taken to be a bound on the essential supremum of \(\sigma(x,r,\Lambda)\), as \(r\) tends to infinity. Geometric objects around each \(v\in\Lambda\) are introduced that encode all of the local geometric information that is needed in the local analysis of the packing. These objects are called decomposition stars, they form a compact topological space DS, and their data are sufficient to determine a Voronoi cell \(\Omega(D)\) for each \(D\in\text{DS}\). Considering a continuous so-called score function \(\delta: \text{D}S\to R\), it is shown that the maximum of \(\delta\) on DS is a certain constant. By this the Kepler conjecture (as an optimization problem in an infinite number of variables, which are the coordinates of the points of \(\Lambda\)) is reduced to an optimization problem in a finite number of variables, it follows a complete characterization of the decomposition stars at which \(\delta\) attains its maximum, yielding the face-centered cubic packing or the hexagonal-close packing, finally confirming the Kepler conjecture. Results on (combinatorial) plane graphs, suitably associated to suitable decomposition stars, play the essential role in difficult proof parts. Various further types of methods are used in this very long and complicated proof. However, the author succeeds in showing the structure of the proof and the interplay between the different tools and methods.

Let \(\Lambda\) denote the set of ball centers in a so-called saturated packing. Denoting by \(B(x,r)\) the closed ball with center \(x\) and radius \(r\), the author introduces \(\sigma(x, r,\Lambda)\) as finite density being the ratio of the volume of \(B(x,r,\Lambda)\) to the volume of \(B(x,r)\) (here \(B(x,r,\Lambda)\) denotes the intersection with \(B(x,r)\) of the union of all balls in the packing). Using Voronoi cells of \(\Lambda\) and certain properties of suitable types of functions, it is shown that there is a constant \(C\) such that for all \(r\geq 1\) and all \(x\in\mathbb R^3\) the inequality \(\sigma(x,r,\Lambda)\leq\pi/\sqrt{18}+C/r\) holds.

Based on this, the precise meaning of Kepler’s conjecture is taken to be a bound on the essential supremum of \(\sigma(x,r,\Lambda)\), as \(r\) tends to infinity. Geometric objects around each \(v\in\Lambda\) are introduced that encode all of the local geometric information that is needed in the local analysis of the packing. These objects are called decomposition stars, they form a compact topological space DS, and their data are sufficient to determine a Voronoi cell \(\Omega(D)\) for each \(D\in\text{DS}\). Considering a continuous so-called score function \(\delta: \text{D}S\to R\), it is shown that the maximum of \(\delta\) on DS is a certain constant. By this the Kepler conjecture (as an optimization problem in an infinite number of variables, which are the coordinates of the points of \(\Lambda\)) is reduced to an optimization problem in a finite number of variables, it follows a complete characterization of the decomposition stars at which \(\delta\) attains its maximum, yielding the face-centered cubic packing or the hexagonal-close packing, finally confirming the Kepler conjecture. Results on (combinatorial) plane graphs, suitably associated to suitable decomposition stars, play the essential role in difficult proof parts. Various further types of methods are used in this very long and complicated proof. However, the author succeeds in showing the structure of the proof and the interplay between the different tools and methods.

Reviewer: Horst Martini (Chemnitz)

### MSC:

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

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

52C35 | Arrangements of points, flats, hyperplanes (aspects of discrete geometry) |