Results 1 to 2 of 2
Like Tree1Thanks
  • 1 Post By emakarov

Thread: maximum distance between point and line

  1. #1
    Senior Member Dinkydoe's Avatar
    Dec 2009

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

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

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

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

    But why must $\displaystyle g(1)$ be this maximum??
    Last edited by Dinkydoe; May 16th 2012 at 04:49 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Oct 2009

    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
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 6
    Last Post: May 16th 2011, 04:57 AM
  2. maximum distance from point to sphere
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: Jan 31st 2011, 04:25 PM
  3. Distance:Line and Point
    Posted in the Trigonometry Forum
    Replies: 2
    Last Post: Apr 26th 2010, 11:36 AM
  4. distance from point to line
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Feb 8th 2009, 02:52 PM
  5. Replies: 3
    Last Post: Feb 20th 2008, 10:17 AM

Search Tags

/mathhelpforum @mathhelpforum