Sphere packings. V: Pentahedral prisms (Q735000)

This paper is the fifth in the series of six papers devoted to the proof of the Kepler's conjecture, all in Discrete Comput. Geom. 36, No. 1. This famous conjecture, open since 1611, asserts that no packing of congruent balls in three dimensions has density greater than the face-centered cubic packing. In the second paper of this series a compact topological space of decomposition stars was defined, and a continuous scoring function on this space was introduced. Moreover, Kepler's conjecture was related to a certain conjecture about location of global maxima of this scoring function. The two conjectured global maxima of the scoring function (corresponding to decomposition stars of the face-centered cubic and hexagonal-close packings) were shown to be local maxima in the third paper of this series. A so-called \textit{contravening} decomposition star is a potential counterexample to the Kepler's conjecture. This paper considers a particular class of potentially contravening decomposition stars, \textit{pentahedral prisms}, on which the scoring function comes very close to its values at the decomposition stars of the face-centered cubic and hexagonal-close packings. The subject of this paper is to prove that in fact pentahedral prisms are not contravening.