# Is this a characterization of Uniformly Convex spaces?

Let $E$ be a Banach space and $\phi:[0,1]\rightarrow E$ a continuous path such that $\phi(0)=0$ and $\phi(1)=a$ with $a\neq 0$. Suppose that $\phi$ is linear by parts, i.e. there exist $t_0=0<t_1<...<t_{n-1},t_n=1$ such that $\phi$ is linear in every interval $[t_{i+1},t_i]$.

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 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 reciprocal true?

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