(in principle) the Minkowski conjecture can be verified in a given dimension using
Appendix A: Polyhedral representation conversion. In Appendix A we
provide some background on polyhedra and in particular on a fundamental problem
in polyhedral combinatorics, on which many computational results described in this
book rely: The representation conversion of polyhedra with large symmetries. We
start with a brief review of some basic properties and explain how to compute
different polyhedral symmetries. We address the group theoretical background
as well as data structures and software tools, which are necessary to deal with
large orbits. Finally, we describe Decomposition methods for the representation
conversion problem, which have proved to perform best in practice on the polyhedra
we treated during our studies.
Conclusions and prospects. In this book we describe foundations and some
applications of a Computational Geometry of Positive Definite Quadratic Forms.
Based on generalizations of Voronoi’s reduction theories, we provide new algorithms
and details on their practical implementation. So far, such implementations have
only been used for applications in the context of lattice sphere coverings (see Chap-
ter 5). However, we are convinced that many more, new results can be obtained.
Some interesting possible future projects are listed in Appendix B.
We think in particular that (in analogy to the lattice covering problem) many
new record breaking lattices for the packing-covering constant and for the so-called
quantizer problem can be found. We strongly believe that the theory described in
this book can be used to find periodic non-lattice sets, which are “better” (with
respect to any of the discussed problems) than any lattice of the same dimension.
We think that it is only a matter of time and a question of suﬃcient computational
resources, until we will see a solution of the 6-dimensional covering and the 9-
dimensional sphere packing problem.
A key ingredient for computational successes described in this book is the
representation conversion of polyhedra under symmetry. We think that further
improvement on the available tools will not only yield further progress for the
problems discussed here, but also for applications beyond the scope of this book.
Furthermore, we are convinced that the future will show more and more mathe-
matical proofs based on rigorous numerical computations and non-linear convex
optimization. However, there is still plenty of groundwork to be done.
So let’s do it. . .