From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: disdance between sets
Date: 10 Dec 1996 20:56:42 GMT
Summary: May I have dis dance?

In article <00002098+0000a7e9@msn.com>,
Douglas Magowan <ferfactor@msn.com> wrote:
>As I remember the definition of distance between two sets A and B is 
>the sum of the distances of every element in set A  from every 
>element in B.
>
>Is this right??
>
>My problem with this is that every set has a distance from its self.

Right -- that would make it a poor definition of a "distance", since
one axiom of a distance function is that it should satisfy  d(x,x)=0
for every  x.

You can define  
	d(A, B) = inf{ d(x,y) : x in  A  and  y in B };
then  d(A, A) = 0,  d(A,B) >= 0 for all  A  and  B, and  d(A,B)=d(B,A).
Those are properties a distance function should satisfy.

If this definition is useful in any particular setting, why, go ahead and
use it. However, there are occasions in which one might prefer something else.

For example, in order to discuss a "distance", one would also like to have  
d(A,B) > 0  if  A  and  B  are distinct.
Clearly this is false (consider what happens if  A  and  B  intersect).
Indeed, one can't even assume  d(A,B) > 0  if  A  and  B  are disjoint
(consider  A = complement of  B), although this _does_ follow if
A  and   B  are compact.

The last axiom for a distance function is the triangle inequality, but
	d(A, B) <= d(A, C) + d(C, B)
is unlikely to be true in any reasonable setting (e.g. if  C   is any
big set, it can be close to both  A  and  B  even if they are far apart.)

However, a distance function _can_ be defined, at least on the collection
of compact sets (in, say, a complete metric space  X). If you define

	d(A, B) = max{ d1(A,B), d1(B,A) }    where
	d1(A,B) = max{ min{ d(x,y) : y in Y } : x in X }

(got that?) then  d  really is a metric, known  as the Hausdorff metric,
and in fact the collection of compact sets becomes a complete metric
space   H(X)  under this metric.

If you unravel this definition, you discover that two sets  A  and  B
are "close" if they are pretty much the same set of points in  X.
This makes it useful for discussing e.g. fractals. For example, the
Koch snowflake is the limit of a certain sequence in  H(R^2).
(The sequence begins with  x_0 = equilateral triangle and continues with
x_(n+1) = x_n union (equilateral triangle resting on each edge of  x_n). )
So you get to use the very concrete ideas of convergence in metric spaces
to discuss these sets.

dave