# maximum distance between point and line

• May 16th 2012, 05:46 AM
Dinkydoe
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??
• May 16th 2012, 08:09 AM
emakarov
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).