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Prove Open Balls in an NLS are convex.

  1. Jun 23, 2011 #1
    Given a Normed Linear Space, prove that all open balls are convex.

    A, a subset of the space is said to be convex if, for all pairs of points (x,y) in a, the point
    z = x + t(y-x) belongs to A. (t goes from 0 to 1).
     
  2. jcsd
  3. Jun 23, 2011 #2

    tiny-tim

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    Hi Oster! :smile:

    Hint: rewrite z as (1-t)x + ty …

    then what can you say about ||z|| ? :wink:
     
  4. Jun 23, 2011 #3
    Hmm

    ||z|| = ||(1-t)x + ty|| <= (1-t)||x|| + t||y|| [norm definition]

    But, if i'm working with a ball centered at 0 with radius r then

    ||x|| and ||y|| would be less than r implying that

    ||z|| < (1-t)r + tr = r

    Which implies z belongs to this ball around 0 with radius r.

    So, all that's left to prove is that any ball B(a,r) = a + B(0,r)?

    Am I correct?
     
  5. Jun 23, 2011 #4

    tiny-tim

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    Hi Oster! :smile:

    (have an leq: ≤ :wink:)
    That's right! :smile:
    you started with the wrong terminology :redface:

    if your two original points are p and q in a ball of centre a, then your x is p - a, and your y is q - a :wink:
     
  6. Jun 23, 2011 #5
    Thanks much!
     
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