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# The Strange World of the Hausdorff Metric Geometry

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# XIX. When Is
A Hausdorff Line Not A Line - 1?

** **

We have discussed many properties
of betweeness in *H*(**R**^{n}),
now we move on to lines. Theorem 5 tells us that, given distinct *A* and *B
*in *H*(**R**^{n}),
there is always at least one point *C *in *H*(**R**^{n}) at each location satisfying *ACB*. Can
we say the same about elements satisfying *ABC* or *CAB*? The
fascinating answer turns out to be no. The reasons are contained in Theorems 3
and 4 of [2], which can be summarized as follows.

**Theorem 7:** *Let A and B be elements of H*(**R**^{n}) *and let* *r*
= *h*(*A*, *B*)*. If d*(*A*, *B*) > *d*(*B*,
*A*) *and *(*A*)_{s}
*is a subset of N*_{r+s}(*B*)

*for some s *> 0, *then there is no element C in H*(**R**^{n}) *satisfying BCA
and h*(*A*, *C*) = *s*_{1} *for any s*_{1}
> *s. *

So if *A* and *B* are
in *H*(**R**^{n})
with

*d*(*A*, *B*) > *d*(*B*, *A*)
*and *(*A*)_{s}
*a subset of N*_{r+s}(*B*) *for some s *> 0

for some *s*,
then there can be no Hausdorff line containing elements farther than *s*
units to the right of *A*. In other words, some Hausdorff lines have holes
in them. Does this ever really happen?

Consider
the example illustrated in the next applet. The set *B* is a two-point set
{*b*_{1}, *b*_{2}} where *b*_{1} = (0, *z*)
and *b*_{2} = (0, -*z*) for some *z* > 0. The set *A*
is also a two-point set {*a*_{1}, *a*_{2}}, where *a*_{1}
is the point at the origin and *a*_{2} = (*x*, *y*) is a
point in the first quadrant with *y* < *z* and *d*_{E}(*a*_{2},
*b*_{2}) > *z*. Then *r*
= *h*(*A*, *B*) = *d*(*A*, *B*) = *d*_{E}(*a*_{2},
*b*_{2}) > *d*(*B*, *A*). The set (*A*)_{s} is colored light
red and (*B*)_{r+s}
is light blue. Note that *N*_{r}_{+s}(*B*) is
the interior of (*B*)_{r+s}.
For some values of *s*, the single point *c* is in (*A*)_{s} but not *N*_{r}_{+s}(*B*).
You can move every point except *a*_{1} to find configurations in
which (*A*)_{s}
is a proper subset of *N*_{r}_{+s}(*B*). In
these examples there is a transition point where (*A*)_{s} is not contained
in *N*_{r}_{+s}(*B*) for values of *s*
less than some *s*_{0} but (*A*)_{s} is a proper subset of *N*_{r}_{+s}(*B*)
for *s* greater than or equal to *s*_{0}. It turns out that
this is always the case and that, in fact, our Hausdorff lines actually become
rays in these situations. In fact, we can construct configurations in which our
rays stop as far to the right of *A* as we like.

How
can we be sure that our Hausdorff lines really are rays? The answer is next.

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