Re: metric in which balls are triangles

In article <1112821803.685824.241000@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
lukasz <bbla32@xxxxx> wrote:
>Can anyone provide a hint on how to define a metric d:R*R->R in which
>all balls are triangles (any or at least isosceles)? I tried with a
>function that returned the circumradius for the isosceles triangle with
>one (bottom) edge parallel to the OX axis, where one coordinate was a
>point on a triangle and the other was the circumcenter; although these
>two points unequivocally defined a triangle, such a function obviously
>did not meet the symmetry condition for a metric.
>
>I suppose the function should rather be similar to the metric in which
>all balls are squares:
>
>d( (x,y), (a,b) ) = max{ |x - a|, |y - b| }
>
>but I can't think of anything like this for triangles. Any ideas?

It's not at all obvious that such a metric can exist. Do you
have any reason to think that it does? Certainly there won't
be a simple formula: it can't be a norm.

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

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