# 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 $\parallel p-q\parallel $.

###### Proof.

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

$$L\ge \parallel q-x\parallel +\parallel x-p\parallel >\parallel p-q\parallel .$$ |

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.
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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 |