I think this has something to do with convexity. The shape of this is like a star and is convex. Is this a necessary condition for a ball?Quote:

Prove that the set is not a unit ball in any metric induced by a norm on .

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- January 28th 2010, 07:55 AMShowcase_22Unit BallQuote:

Prove that the set is not a unit ball in any metric induced by a norm on .

- January 28th 2010, 02:22 PMDrexel28
- January 28th 2010, 10:41 PMShowcase_22

So it could never be a norm!

Returning to the idea of convexity, does convexity have any relation to a set being an open ball or not?

Thankyou! - January 29th 2010, 07:00 AMDrexel28
- January 29th 2010, 07:04 AMShowcase_22
Yes.

I was talking to one of my friends today and he said that open ball Convexity. No other properties link the two.

Is this true? It's just that it would be very helpful if there was a stronger statement that could be used! - January 29th 2010, 07:07 AMDrexel28
My point is that the way you defined convexity can only be determined (visually since that is how you are describing it) in . If I asked you a similar question for the Banach space where is some arbitrary metric (or even topological) space I doubt you could graph it and tell me whether or not it's "convex"