The Strange World of the Hausdorff Metric Geometry

IX. Metric Lines, Halflines, and Segments

As we have seen, there can be more than one point at each location satisfying various versions of the triangle equality. This makes it awkward to define a line in H(Rⁿ) as the collection of elements satisfying (1). It might be more natural to think of a line (or a segment) as a distance preserving image (a distance preserving map is called an isometry) of a Euclidean line (or segment). This leads to the following definitions as in [5, 6].

Definition: An isometry between metric spaces (X, d_X) and (Y, d_Y) is a function f: X → Y so that

d_Y(f(a), f(b)) = d_X(a, b)

for all a, b in X.

Definition: Let X be a metric space with metric d.

1. A subset S of X is a metric segment with endpoints a ≠ b in X if S is isometric to the interval [0, d(a, b)] in R .

2. A subset L of X is a metric line (or halfline) if there is an isometric embedding f: R → X (f: (0, ∞) → X) so that

f(R) = L (f((0, ∞)) = L).

In the next sections we will see theorems and examples that tell us about segments, lines, and halflines in H(Rⁿ).