Lecture 4 33
In general, given a compact metric space X, for any ε 0, let
N(ε) be the minimum number of ε-balls required to cover X; that
is, the minimum number of points x1, . . . , xN(ε) in X such that every
point in X lies within ε of some xi. This may be thought of as
measuring the average ‘volume’ of an ε-ball, in some sense; the upper
box dimension of X is defined to be
(X) = lim sup
log N(ε)
log 1/ε
We take the upper limit because the limit itself may not exist. The
lower box dimension is defined similarly, taking the lower limit in-
stead. These notions of dimension do not behave quite so nicely as
we would like in all situations; for example, the set of rational num-
bers, which is countable, has upper and lower box dimension equal to
There is a more effective notion of Hausdorff dimension, which
eliminates the need to distinguish between upper and lower limits,
and which is equal to zero for any countable set; because its definition
requires an understanding of measure theory, we will not discuss it
here. For ‘good’ sets all three definitions coincide, and are central
to the study of fractal geometry; however, they are not topological
invariants, so our claim in the last section must be understood to
apply only to a strictly topological notion of dimension.
d. Geodesics. When we are interested in a metric space as a geo-
metric object, rather than as something in analysis or topology, it is
of particular interest to examine those triples (x, y, z) for which the
triangle inequality becomes degenerate, that is, for which d(x, z) =
d(x, y) + d(y, z).
For example, if our space X is just the Euclidean plane R2 with
distance function given by Pythagoras’ formula,
d((x1, x2), (y1, y2)) = (y1 x1)2 + (y2 x2)2,
then the triangle inequality is a consequence of the Cauchy-Schwarz
inequality, and we have equality in the one iff we have equality in the
other; this occurs iff y lies in the line segment [x, z], so that the three
points x, y, z are in fact collinear.
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