From: Stephen Montgomery-Smith <stephen@math.missouri.edu>
Subject: Re: taxicab geometry
Date: Wed, 20 Dec 2000 03:15:42 GMT
Newsgroups: sci.math
Summary: [missing]

The taxicab distance is more usually called the l_1 (ell sub 1)
distance, as part of a large class of distances l_p for 
0<p<=infinity.  The usual Euclidean distance is l_2.  (l_p does
not satisfy the triangle inequality if p<1.)

There is a huge huge literature on these as part of the larger
study of Banach spaces.  There are many beautiful results.
For example, consider 2n dimensional space with the l_1 distance
(usually denoted l_1^{2n}).  Then there is a n-dimensional subspace
E such that the l_1 distance restricted to E is almost isormorphic
to Euclidean distance, that is the ratio of the l_1 distance and the
l_2 distance on E is almost a constant.  This is part of the 
celebrated result of a famous chap whose name I remember but not
the spelling of his name (Kachine or something like that).
The beauty of this result is that the existence of the subspace
is proved using random methods (that is it is shown that if a
subspace is chosen randomly then with probability close to 1 
it has the desired property).  But no explicit construction
of this subspace is known.


John Prussing wrote:
> 
> Is taxicab geometry typically used as a example in non-Euclidean geometry,
> or is it considered more of *popular* curiosity?  At least on the
> popular side "pi" = 4 in taxicab space gets people to think about what
> a "circle" is, what "circumference" is, essentially what distance is,
> in a non-Euclidean space.
> 
> If it is not used, what is it lacking?
> --
> =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
> John E. Prussing
> Dept. of Aeronautical & Astronautical Engineering
> University of Illinois at Urbana-Champaign
> http://www.uiuc.edu/~prussing
> =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

-- 
Stephen Montgomery-Smith
stephen@math.missouri.edu
http://www.math.missouri.edu/~stephen