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, R^{d}; in particular, R^{1} = R is just the set of real
numbers or the real line, R^{2} is the
(Euclidean) plane, and R^{3} is the usual
(Euclidean) space. Points in R^{d} are printed
in bold type x, y,
and so on, and we will sometimes use the coordinate form x = (x_{1}, …, _{λ}x_{d}). If x and
y are points of R^{d}, the distance between them
is .

Sets, which will generally be subsets of R^{d}, are denoted by capital letters (e.g., E, F, and K). In the usual way, x ∈ E means that the point x
is a member of the set E, and E ⊂ F 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 B_{r}(x) =
{y:|y − x| ≤ r}. Similarly, the
open ball is {y:|y − x| < r}. Thus, the closed ball contains its bounding sphere,
but the open ball does not. Of course, in R^{2},
a ball is a disk, and in R^{1}, a ball is just
an interval. If a < b, we write [a, b]
for the closed interval {x:a ≤ x ≤ b} and (a, b) for the open interval {x:a < x < b}.

We write E ∪ F for the union of the sets E