Let $E$ be a Banach space and let $\phi\colon [0,1]\rightarrow E$ be a continuous path such that $\phi(0)=0$ and $\phi(1)=a$ with $a\neq 0$. Suppose that $\phi$ is piecewise linear, i.e., there exist $t_0=0
Define the length of $\phi$ by $l(\phi)=\displaystyle\sum_{i=0}^{n-1}\|\phi(t_{i+1})-\phi(t_i)\|.$ Suppose that $E$ is uniformly convex. Is it true that the only path linear by parts joining $0$ and $a$ with length $l(\phi)=\|a\|$ is the path $\phi(t)=at$ ? Is the converse true?