Length Spaces
In this section we consider metric spaces in which the distance between two points is given by the infimum of the lengths of curves which join them — such a space is called a length space. In this context, it is natural to allow metrics for which the dist
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In this section we consider metric spaces in which the distance between two points is given by the infimum of the lengths of curves which join them - such a space is called a length space. In this context, it is natural to allow metrics for which the distance between two points may be infinite. A convenient way to describe this is to introduce the notation [0, 00] for the ordered set obtained by adjoining the symbol 00 to the set of non-negative reals and decreeing that 00 > a for all real numbers a. We also make the convention that a + 00 = 00 for all a E [0,00]. Having made this convention, we can define a (generalized) metric on a set X to be a map d : X x X ~ [0,00] satisfying the axioms stated in (1.1). Henceforth we shall allow metrics and pseudometrics to take the value
00.
Length Metrics 3.1 Definition. Let (X, d) be a metric space. d is said to be a length metric (otherwise known as an inner metric) if the distance between every pair of points x, y E X is equal to the infimum of the length of rectifiable curves joining them. (If there are no such curves then d(x, y) = 00.) If d is a length metric then (X, d) is called a length space.
°
A complete metric space X is a length space if and only if it has approximate midpoints in the sense that for all x, y E X and e > there exists z E X such that max{d(x, z), d(z, y)} ::: e + d(x, y)j2. An arbitrary metric space gives rise to a length space in an obvious way: 3.2 Proposition. Let (X, d) be a metric space, and let d : X x X ~ [0,00] be the map which assigns to each pair of points x, y E X the infimum of the lengths of rectifiable curves which join them. (If there are no such curves then d(x, y) = 00.) (I)
d is a metric.
(2)
d(x, y)
(3)
If c : [a, b]
~
d(x, y)for all x, y
E
X.
~ X is continuous with respect to the topology induced by d, then
it is continuous with respect to the topology induced by d. (The converse is false in general.) M. R. Bridson et al., Metric Spaces of Non-Positive Curvature © Springer-Verlag Berlin Heidelberg 1999
Length Metrics
33
(4)
If a map e
(5)
The length of a curve e : [a, b] ~ X in (X, d) is the same as its length in (X, d).
(6)
d = d.
: [a, b] ~ X is a rectifiable curve in (X, d), then it is a continuous and rectifiable curve in (X, d).
Proof. Properties (1) and (2) are immediate from the definition oflength. (2) implies (3), and (6) is a consequence of (4) and (5). Property (4) is a consequence of 1.20(5), so it only remains to prove (5). Let e : [a, b] ~ X be a path which has length lee) with respect to the metric d, and length lee) with respect to the metric d. On the one hand we have that lee) ::: l(c), by (2), and on the other hand k-l
l(c) =
sup
Ld(e(ti), e(ti+l»
a=to::::Jl::;···::;tk=b i=O
k-l
~
sup
L
l(el[tH.t;]) = lee).
a=to::::Jl::;···::;tk=b i=O
o
Hence l(c) = lee).
3.3 Definition. Let (X, d) be a metric space. The map d defined in (3.2) is called the length metric (or inner metric) associated to d,.and (X, d) is called the length space associated to (X, d). Note that d = d if an o
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