In this chapter we review some topics that are needed in the main part of the text. The book uses tools from various branches of mathematics related to convexity; hence, to give the reader a chance to follow the presentation without constantly checking references, this introductory part is more extensive than usual. We present only a few proofs where the exact statement we need is not so easy to access. The chapter is organized in a way that most of the material needed in Part 1 is contained in Sections A.1–A.6.

Before going into details we define the central notions of this book. A set K is called convex if it contains any segment whose endpoints lie in K. In addition, K is a convex body if K is compact and its interior is nonempty. Planar convex bodies are also known as convex domains. A family {Kn} of convex bodies is called a packing if the interiors of any two Ki and Kj, i ≠ j, are disjoint, or, in other words, if Ki and Kj do not overlap. Next, {Kn} is a covering of a set X if the union of the convex bodies contains X. Finally, {Kn} is a tiling of X if each Kn is contained in X, and {Kn} is both a packing and a covering of X.

Some General Notions

As usual ℕ, ℤ, ℝ and denote the family of natural numbers, integers, and real numbers, respectively.