Show that if $\sigma$ is a trajectory between two points $p$ and $q$ and $L(\sigma)$ its arc length, then $L(\sigma)=||p-q|| \iff \sigma$ is a straight line.
Its easy to prove, with $\sigma(a)=p$, $\sigma(a)=q$ and using the definition $L(\sigma)=\int^{a}_{b}||\sigma'(t)||dt$ that if $\sigma$ is a straight line then $L(\sigma)=||p-q||$, but how do I prove the converse?