Appendix: Background
Published online by Cambridge University Press: 12 September 2009
Summary
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.
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- Information
- Finite Packing and Covering , pp. 325 - 356Publisher: Cambridge University PressPrint publication year: 2004