# 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 StraightLineIsShortestCurveBetweenTwoPoints 2013-03-22 15:39:43 2013-03-22 15:39:43 stevecheng (10074) stevecheng (10074) 11 stevecheng (10074) Result msc 51N05 ArcLength Rectifiable