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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 such that $\phi$ is linear on 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 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?

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For a normed space are equivalent:

i) the only geodesic between any pair of two points is the affine segment.

ii) the space is strictly convex.

See Prop. 7.2.1 on p.180 in Metric Spaces, Convexity and Nonpositive Curvature by A. Papadopoulos for this equivalence and quite a few others.

On the other hand, Day showed that there are (reflexive) strictly convex spaces which are not even isomorphic to a uniformly convex space, so the converse you ask about is not true (unless the space is finite-dimensional).