straight line is shortest curve between two points

Suppose $p$ and $q$ are two distinct points in $\mathbb{R}^{n}$ , and $\gamma$ is a rectifiable curve from $p$ to $q$ . Then every curve other than the straight line segment from $p$ to $q$ has a length greater than the Euclidean distance $\lVert p-q\rVert$ .

Proof.

Let $\gamma\colon[0,1]\to\mathbb{R}^{n}$ be the curve with length $L$ . If it is not straight¹¹If $\gamma$ is a straight line segment but is not injective, that is, it moves $p$ and $q$ , then it is obvious that $L>\lVert p-q\rVert$ ., then there exists a point $x=\gamma(t)$ that does not lie on the line segment from $p$ to $q$ . We have

$L\geq\lVert q-x\rVert+\lVert x-p\rVert>\lVert p-q\rVert\,.$

The first inequality comes from the definition of $L$ as the least upper bound of the length of any broken-line approximation to the curve $\gamma$ . The second inequality is the usual triangle inequality, but it is a strict inequality since $x$ lies outside the line segment between $p$ and $q$ , as shown in the following diagram. ∎

Title	straight line is shortest curve between two points
Canonical name	StraightLineIsShortestCurveBetweenTwoPoints
Date of creation	2013-03-22 15:39:43
Last modified on	2013-03-22 15:39:43
Owner	stevecheng (10074)
Last modified by	stevecheng (10074)
Numerical id	11
Author	stevecheng (10074)
Entry type	Result
Classification	msc 51N05
Related topic	ArcLength
Related topic	Rectifiable