Unit Ball

**Showcase_22** · January 28th 2010, 07:55 AM

Prove that the set $G=\{(x_1,x_2) \ : |x_1|^{\frac{1}{3}}+|x_2|^{\frac{1}{3}} \leq 1 \}$ is not a unit ball in any metric induced by a norm on $\mathbb{R}^2$ .

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?

**Drexel28** · January 28th 2010, 02:22 PM

Originally Posted by Showcase_22

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?

Forget the open ball for a second. What characteristic of a norm would be violated by this picture?

**Showcase_22** · January 28th 2010, 10:41 PM

$||\lambda x||=|\lambda x_1|^{\frac{1}{3}}+|\lambda x_2||^{\frac{1}{3}} \neq \lambda |||x||$

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!

**Drexel28** · January 29th 2010, 07:00 AM

Originally Posted by Showcase_22

$||\lambda x||=|\lambda x_1|^{\frac{1}{3}}+|\lambda x_2||^{\frac{1}{3}} \neq \lambda |||x||$

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!

What does convexity mean in this sense? Do you mean that the ball has "dips" in it?

**Showcase_22** · January 29th 2010, 07:04 AM

Yes.

I was talking to one of my friends today and he said that open ball $\Rightarrow$ 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!

**Drexel28** · January 29th 2010, 07:07 AM

Originally Posted by Showcase_22

Yes.

I was talking to one of my friends today and he said that open ball $\Rightarrow$ 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!

My point is that the way you defined convexity can only be determined (visually since that is how you are describing it) in $\mathbb{R}^2,\mathbb{R}^3$ . If I asked you a similar question for the Banach space $\mathcal{C}\left[X,\mathbb{C}\right]$ where $X$ is some arbitrary metric (or even topological) space I doubt you could graph it and tell me whether or not it's "convex"

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