minkowski

Dave Keenan wrote:

>I agree that bringing Euclid in confuses matters, but so does Minkowski. I
don't know how Euclid *or* Minkowski got in here at all! Both Euclidean and
Minkowski distances relate to *geometries*, i.e. continuous (or
approximately continuous) spaces. They have nothing to do with distances on
graphs or lattices (which are definitely discrete, even when the edges are
weighted).

Dave, there are other uses of the Minkowski metric.
I refer you to Rudin's book Real and Complex analysis,
page 65, where the inequality

( integral (f+g)^p )^(1/p) <= (integral f^p )^(1/p) + (integral g^p)^(1/p )

(sorry about the asci math)
is defined as "Minkowski's inequality". This is used
to show that distance measures with various values of p
are norms. In particular, p=1 (interpret f^1 as absolute value) is the "city block metric" p=1/2, is the euclidean distance, etc.
These work whether its a continuous space or a discrete space
(see page 67, in which case replace the integral by a sum).

Basically, the result is that any value of p>=1 gives a well defined
norm, and hence might be a candidate for a distance metric in these discussions.