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Math Help - maximum distance between point and line

  1. #1
    Senior Member Dinkydoe's Avatar
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    maximum distance between point and line

    Hoi, I'm trying to prove something seemingly trivial...:PP

    But i can't prove the following for Banach-spaces...i thought of many ways trying to solve this:

    consider a line, say  l(t) = t(y-x)+x with t\in [0,1]...and a point a

    I want to show that the maximal distance to a, (not on the line) and a point on the line l(t), is between a, and one of
    the boundary points of the line l(0) or l(1). That is, consider g(t) = \left\|l(t)-a\right\| and assume g(0)\leq g(1).

    How can I show this (clearly continuous) function attains a maximum on g(1). I cant prove this seemingly easy question.
    I just don't see how this goes wrong when I assume that g attains a maximum for any s\in (0,1).

    [0,1] is compact, therefore g must attain a maximum and a minimum somewhere....

    But why must g(1) be this maximum??
    Last edited by Dinkydoe; May 16th 2012 at 04:49 AM.
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  2. #2
    MHF Contributor
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    Re: maximum distance between point and line

    This can be solved considering a triangle and a segment joining a vertex with the opposite side. We have a - l(t) = (1 - t)(a - l(0)) + t(a - l(1)), so g(t) <= (1 - t)g(0) + tg(1) <= [since g(0) <= g(1)] (1 - t)g(1) + tg(1) = g(1).
    Thanks from Dinkydoe
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