Thread: 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 with ...and a point

I want to show that the maximal distance to , (not on the line) and a point on the line , is between , and one of
the boundary points of the line or . That is, consider and assume .

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

is compact, therefore must attain a maximum and a minimum somewhere....

But why must be this maximum??

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).