Gruber P. Convex and Discrete Geometry

Geometry of Numbers and Aspects of Discrete
Geometry
The roots of discrete geometry and geometry of numbers date back to the seventeenth
or eighteenth century. We mention the ball packing problem, first treated by Kepler.
The well known discussion between Newton and Gregory on the problem of the
13 balls involved the following question: given a unit ball, is it possible to arrange 13
non-overlapping unit balls such that each touches the given ball, or not? Gregory
said yes, Newton no. Sporadic results in the late eighteenth and the nineteenth century
are due to Lagrange, Gauss, Dirichlet, Korkin and Zolotarev (packing of balls
and positive definite quadratic forms), Fedorov (tiling), and Thue (irregular packing
of circular discs). Both areas became well-established branches of mathematics only
at the turn of the nineteenth and during the twentieth century. The major figures at
the beginning of the systematic era were Minkowski (fundamental theorems, applications
to Diophantine approximation) and Voronoĭ (geometric theory of quadratic
forms) in the geometry of numbers and, 50 years later, Fejes Tóth (packing and
covering) in discrete geometry. Other contributors to both areas in the twentieth century
were Delone, Siegel, Mahler, Davenport, Kneser, Rogers, Ryshkov and many
living mathematicians, including Hlawka, Bambah and Schmidt. Important topics
are lattice and non-lattice packing, covering and tiling. Both areas have strong ties to
other parts of mathematics and the applied sciences, for example to crystallography,
coding and data transmission, modular functions, computational and algorithmic
geometry, graph theory, number theory and algebraic geometry. There are applications
to numerical integration and the Riemann mapping theorem.
At the heart of the geometry of numbers is the interplay of the group-theoretic
notion of lattice and the geometric concept of convex set, the lattices representing
periodicity, the convex sets geometry. In discrete geometry similar problems are considered
as in the geometry of numbers, but relaxing periodicity.
While the important problems of the geometry of numbers and of discrete
geometry are easy to state, their solution, in general, is difficult. Thus progress is
slow. Major results in recent years are at the boundary of the classical theory, dealing,
for example, with positive and indefinite quadratic forms and computational and
algorithmic aspects. It seems that fundamental advance in the future will require new
ideas and additional tools from other areas.