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Re: Balance

John Pryce schrieb:


Ulrich Kulisch wrote:

4. Can anybody tell me whether the Hausdorff metric can > be defined for extended intervals?


Good question and I would like other people's views on this.

For subsets A,B of a general metric space X with distance d(x,y), the definition I know is
  D(A,B) = larger of (
    sup over a in A of d(a,B)
    sup over b in B of d(b,A)
    ),
where for x in X and a subset Y of X
  d(x,Y) = inf over y in Y of d(x,y).

So for _arbitrary_ nonempty A,B, D(A,B) is a non-negative real, or +oo.
The usual conventions inf(Empty) = +oo, sup(Empty) = -oo give unambiguous values also when one or both of A,B are empty.


I prefer d(X,Y) = NaN when X or Y is empty.
In an extended-real-based system, the usual metric does _not extend_ to infinite points, so that to my mind it makes no sense to say, for instance, d(+oo,+oo) = 0. 

Why not?

d(x,x)=0 is a very natural requirement for any distance.


Arnold Neumaier