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Math Help - Unit Ball

  1. #1
    Super Member Showcase_22's Avatar
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    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?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Showcase_22 View Post
    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?
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  3. #3
    Super Member Showcase_22's Avatar
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    ||\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!
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Showcase_22 View Post
    ||\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?
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  5. #5
    Super Member Showcase_22's Avatar
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    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!
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Showcase_22 View Post
    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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