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) > > >Sent via Deja.com >http://www.deja.com/