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)


 * For $x_1 = x_2$ specifically, we have the distinct issue that $f$ is not a function, and thus also not differentiable. I guess rotating the plane would be easiest, so we can just take the function in terms of $y$ instead of $x$. The hardest part is dealing with continuous functions that are not differentiable, in general, is showing that it will have a very long arc length. In general, curves that are continuous but differentiable nowhere or not almost everywhere are not rectifiable (see [this]), but for curves which are differentiable almost everywhere, the points of where it isn't differentiable affect the curve's length infinitesimally. --PeterJohnson (talk) 15:18, 26 March 2023 (UTC)


 * What I mean, more formally (for the latter section on differentiability), is that if the set of points where $f(x)$ is not differentiable is not finite, it has infinite curve length. Clearly, if the length is infinite, it smaller than any finite curve, including the line. If the set of points where $f(x)$ is not differentiable is finite, then they have an infinitesimal (aka 0 for our purposes in $\R$) effect on the length. My only concern is whether the tools requisite to formalize such an idea should be presented as a Lemma or a seperate proof. A proof of this would be rather long to set up, and also rather tedious. I feel it deserves a new page. --PeterJohnson (talk) 15:29, 26 March 2023 (UTC)