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[parent] straight line is shortest curve between two points (Result)

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 straight 1, then there exists a point $ x = \gamma(t)$ that does not lie on the line segment from $ p$ to $ q$. We have

$\displaystyle 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. $ \qedsymbol$
\includegraphics{curve-triangle.eps}


Footnotes

... straight1
If $ \gamma$ is a straight line segment but is not injective, that is, it moves back and forth between $ p$ and $ q$, then it is obvious that $ L > \lVert p-q\rVert $.


"straight line is shortest curve between two points" is owned by stevecheng.
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See Also: arc length, rectifiable curve


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Cross-references: diagram, strict inequality, triangle inequality, approximation, least upper bound, inequality, lie on, obvious, injective, Euclidean distance, line segment, straight, curve, rectifiable curve, points

This is version 8 of straight line is shortest curve between two points, born on 2006-02-06, modified 2006-09-04.
Object id is 7595, canonical name is StraightLineIsShortestCurveBetweenTwoPoints.
Accessed 1851 times total.

Classification:
AMS MSC: 51N05 (Geometry :: Analytic and descriptive geometry :: Descriptive geometry)
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