1. ## Unit Ball

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?

2. 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?

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

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!

4. 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?

5. 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!

6. 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"