This monograph gives a short introduction to the relevant modern parts
of discrete geometry, in addition to leading the reader to the frontiers
of geometric research on sphere arrangements. The readership is aimed at
advanced undergraduate and early graduate students, as well as
interested researchers. It contains more than 40 open research problems
ideal for graduate students and researchers in mathematics and computer
science. Additionally, this book may be considered ideal for a
one-semester advanced undergraduate or graduate level course.
The core part of this book is based on three lectures given by the
author at the Fields Institute during the thematic program on "Discrete
Geometry and Applications" and contains four core topics. The first two
topics surround active areas that have been outstanding from the birth
of discrete geometry, namely dense sphere packings and tilings. Sphere
packings and tilings have a very strong connection to number theory,
coding, groups, and mathematical programming. Extending the tradition of
studying packings of spheres, is the investigation of the monotonicity
of volume under contractions of arbitrary arrangements of spheres. The
third major topic of this book can be found under the sections on
ball-polyhedra that study the possibility of extending the theory of
convex polytopes to the family of intersections of congruent balls. This
section of the text is connected in many ways to the above-mentioned
major topics and it is also connected to some other important research
areas as the one on coverings by planks (with close ties to geometric
analysis). This fourth core topic is discussed under covering balls by
cylinders.
