Re: Balance
Dear Ulrich and P1788 members
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.
So in a real-based system (taking the usual metric |x-y|)the answer is Yes, and for instance I make D([5,+oo], [8,+oo]) = 3.
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. Hence, in my BIAS implementation, I have set D(A,B) = NaN if A and B have a common infinite endpoint, eg.
D([5,+oo], [8,+oo]) = NaN.
The set R* of extended reals is of course metrisable: one can map it to a finite interval by, say, x -> tanh(x) and thus take Hausdorff distance w.r.t. d(x,y) = |tanh(x) - tanh(y)|, but that is not "natural".
Do people agree with this approach?
John Pryce
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