From: ullrich@math.okstate.edu (David C. Ullrich)
Subject: Re: metric problem
Date: Fri, 26 Jan 2001 15:07:10 GMT
Newsgroups: sci.math
Summary: Metrics on R^n for which the balls are not convex

On Fri, 26 Jan 2001 11:52:15 GMT, brianwallace@my-deja.com wrote:

>does there exist a metric on a real linear space which produces non
>convex balls?

Well of course there does - you didn't ask for any connection
between the metric and the vector-space structure or any
topology on the vector space or anything. There is a metric
on R such that all the balls are _dense_ in the standard
topology (not the topology induced by the metric).

What you meant to ask might be either "is there a metric
on R^n which induces the standard topology and which
has non-convex balls", or you might have meant to
ask "is there a metric on R^2 making R^2 into a 
topological vector space for which the balls are
non-convex". The answer to these questions is not
"of course" as with the question you asked, but
it's still yes:

You can define a metric on R^2 by

d((x,y), (s,t)) = |x-s|^(1/2) + |y-t|^(1/2) .

That's a metric with non-convex balls that induces
the standard topology on R^2.

>(balls are also known as open spheres or spheres)
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