straight line is shortest curve between two points

Suppose p and q are two distinct points in n, and γ 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 p-q.


Let γ:[0,1]n be the curve with length L. If it is not straight11If γ is a straight line segment but is not injectivePlanetmathPlanetmath, that is, it moves p and q, then it is obvious that L>p-q., then there exists a point x=γ(t) that does not lie on the line segmentMathworldPlanetmath from p to q. We have


The first inequalityMathworldPlanetmath comes from the definition of L as the least upper bound of the length of any broken-line approximation to the curve γ. The second inequality is the usual triangle inequalityMathworldMathworldPlanetmath, 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