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Homework Help: Euclidean spaces

  1. Jan 18, 2010 #1
    1. The problem statement, all variables and given/known data
    This is a problem from baby rudin in Chapter 1, I've done all of them except this(problem 16).
    suppose k>=3,x,y belongs to [R][/k],|x-y|=d>0,and r>0.prove:
    (1)if 2r>d, there are infinitely many z belong to R(k) such that |z-x|=|z-y|=r.
    (2)if 2r=d, there is exactly one such z.
    (3)if 2r<d, there is no such z.

    3. The attempt at a solution
    I have done a part of this question. I can't prove (1). About (2), when 2r=d, let z=(x+y)/2, then |z-x|=|z-y|=d/2. However, I can't prove it is unique. About (3), suppose there exists such z, then by thereom 1.37 in baby rudin we have 2r=|z-x|+|z-y|>=|x-y|=d, which contradicts the fact that 2r<d.

    May somebody help me!
     
  2. jcsd
  3. Jan 18, 2010 #2
    OMG,will somebody tell me why there is not anybody replied?
     
  4. Jan 18, 2010 #3

    Mark44

    Staff: Mentor

    If someone tells you why no one has replied, then someone would have replied, making the need for an answer to your question unnecessary.:biggrin:

    For the three parts of your question, you are dealing with three points in Rk: x, y, and z.

    Draw a picture of x and y at a distance of d apart. Now add a point z. For part 2, it's pretty clear from the geometry that there is only one place that z could be. For part 3, it's also pretty clear that there could not be any point z that satisfies the given condition.

    For part 1, the geometry shows that there are an infinite number of possible locations for z.
     
  5. Jan 18, 2010 #4
    For the three parts of your question, you are dealing with three points in Rk: x, y, and z.

    Draw a picture of x and y at a distance of d apart. Now add a point z. For part 2, it's pretty clear from the geometry that there is only one place that z could be. For part 3, it's also pretty clear that there could not be any point z that satisfies the given condition.

    For part 1, the geometry shows that there are an infinite number of possible locations for z.[/QUOTE]
    I appreciate it very much. However, I don't think it is an analytic way. Any way, thank you for so kind to a noob.
     
  6. Jan 18, 2010 #5

    Mark44

    Staff: Mentor

    But drawing a sketch shows you the geometry, which should get you thinking about how you would write up a more analytical explanation.
     
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