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Math Help - easy (?) intersection question

  1. #1
    Senior Member MacstersUndead's Avatar
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    easy (?) intersection question

    any help will be appreciated for this problem. I have an idea what the solution should be, but I'm unsure if I have enough justification.
    --

    Let p = (0,1) and q = (0,-1). Let X be the subset of R^2 that is formed by intersecting all the closed Euclidean balls that contain p and q. What is X?

    Solution (so far):

    We can immediately exclude closed Euclidean balls that do not have p and q as a boundary point, since there exist Euclidean balls contained in them, and that if X is contained in Y, the intersection is X.

    With some basic geometry, I've shown that there are three balls that have p and q as boundary points: the unit ball centered at the origin, and balls of radius sqrt(2) centered at (-1,0) and (1,0).

    Now, I want to say that the intersection between the last two balls mentioned is X, but I cannot convince myself that is the case. Am I doing something wrong?
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  2. #2
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    Quote Originally Posted by MacstersUndead View Post
    Let p = (0,1) and q = (0,-1). Let X be the subset of R^2 that is formed by intersecting all the closed Euclidean balls that contain p and q. What is X?
    Look d(p,q)=\sqrt{2}>0 so let \delta=\frac{d(p,q)}{4}.

    Here are two closed balls: \overline{\mathcal{B}\left(p;\delta\right)}~\&~\ov  erline{\mathcal{B}\left(q;\delta\right)}.

    Can they intersect?
    Last edited by Plato; February 9th 2011 at 03:55 PM.
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  3. #3
    Senior Member MacstersUndead's Avatar
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    d(p,q) = 2, but a simple mistake that doesn't lead to any trouble.

    I understand the direction that you are taking, using a sufficiently small delta so that the two closed balls don't intersect, but they don't contain p and q.
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    You are right. I misread it as 1,0)~\&~q0,-1)" alt="p1,0)~\&~q0,-1)" />.
    But none the less there are two disjoint closed balls centered at p&q. Their intersection is empty.
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  5. #5
    Senior Member MacstersUndead's Avatar
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    Thank you for your help, but I'm still confused. If both Euclidean closed balls contain p and q, then the intersection of those balls would at the very least contain the points p and q.
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    You mean that these balls each contain both p & q .
    Does the problem really say just that.
    Last edited by Plato; February 10th 2011 at 03:58 AM.
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    That is what was said: " intersecting all the closed Euclidean balls that contain p and q."

    If it had said " intersecting all the closed Euclidean balls that contain p or q", then I would have interpreted it the way you did. In this case, the intersection is the Euclidean line segment having p and q as endpoints.
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  8. #8
    Senior Member MacstersUndead's Avatar
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    Oh! *facepalm* I apologize for the confusion. I interpreted that they contain p or q. I will ask for clarification on the question today. Thank you both.
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