Talk:Straight Line Segment is Shortest Path between Two Points

Changing the Title
When I created the page, it should've said "A Line Segment is the Shortest Path Between Two Points" but instead it says "A Line Segment is the Path Between Two Points!" Can we fix it? :(


 * Done. Better to craft this such that the statement is independent of the cartesian plane in which the points are embedded, and introduce its embedding in the cartesian plane as the first part of the proof. --prime mover (talk) 20:14, 25 March 2023 (UTC)
 * Would this truly be any more general? Homeomorphisms aren't length-preserving in the informal sense. --PeterJohnson (talk) 20:53, 25 March 2023 (UTC)


 * It doesn't make it more "general" as such, it just couches the formula in the language of pure geometry before you get round to imposing a coordinate frame on it.


 * Otherwise you would be unable to use the result in a plane without the annoyance of having to embed that plane in a Cartesian frame.


 * And please sign your posts. --prime mover (talk) 08:37, 26 March 2023 (UTC)

Proof holes
a) What if $x_1 = x_2$? Then the argument cannot be applied.

The proof can be salvaged by rotating the plane and making use of the fact that a rotation is an isometry. But then when you bring that machinery into play, you might as well rotate and translate the plane so that $A$ is $\tuple {0, 0}$ and $B$ is $\tuple {a, 0}$ for whatever $a$ is appropriate. Then a lot of the complications go away.

b) What if the function $f$ passing through both $A$ to $B$ is not differentiable? Such functions exist, and for such a function this argument is invalid. --prime mover (talk) 08:37, 26 March 2023 (UTC)