Thread: Integral Inequality

I am trying to prove that the shortest distance between two points is a straight line. And it is almost complete, if I can use the following theorem.
Assume,
are Integratable on .
And,

Then is it true that,
?

And one more,

Und,

Then,


I tried the Cauchy-Swarthz inequality because this is an inner product space but it does not help sufficiently.

Hey PH. Here's a nice proof if you wanna check it out. Something you may like.

Proof shortest distance between two point is a straight line

Originally Posted by galactus

Hey PH. Here's a nice proof if you wanna check it out. Something you may like.

Proof shortest distance between two point is a straight line

Well, I did it with my professor. The proof is so much nicer. This happens to be a Calculus of Variations problem. He told me the standard way is to define "what you think is the answer is" in this case the line passing through the two points. And let be any other path taken, where is some smooth function. And then show the arc length of is more than the arc length of . I was able to do it by the use of the inequalities above. And if someone says they are valid (and I think they are) I can post my solution.