Appendix

A1.1 MATHEMATICAL PRELIMINARIES

We shall require some definitions and notation from set theory. Most of our problems are posed in d-dimensional Euclidean space, Rd; in particular, R1 = R is just the set of real numbers or the real line, R2 is the (Euclidean) plane, and R3 is the usual (Euclidean) space. Points in Rd are printed in bold type x, y, and so on, and we will sometimes use the coordinate form x = (x1, …, λxd). If x and y are points of Rd, the distance between them is .

Sets, which will generally be subsets of Rd, are denoted by capital letters (e.g., E, F, and K). In the usual way, xE means that the point x is a member of the set E, and EF means that E is a subset of F. We write {x: condition} for the set of x for which the condition is true. The empty set, which contains no elements, is written as Ø. We sometimes use a superscript, +, to denote the positive elements of a set (e.g., R+ is the set of positive real numbers).

The closed ball of center x and radius r is defined by Br(x) = {y:|yx| ≤ r}. Similarly, the open ball is {y:|yx| < r}. Thus, the closed ball contains its bounding sphere, but the open ball does not. Of course, in R2, a ball is a disk, and in R1, a ball is just an interval. If a < b, we write [a, b] for the closed interval {x:axb} and (a, b) for the open interval {x:a < x < b}.

We write EF for the union of the sets E

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