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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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up vote 1 down vote accepted

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).

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