From Wikepedia, the definition of geodesic is stated as:

A curve $\gamma: I\to M$ from an interval $I$ of the reals to the metric space $M$ is a geodesic if there is a constant $v\geq 0$ such that for any $t\in I$ there is a neighborhood $J$ of $t$ in $I$ such that for any $t_1, t_2 \in J$ we have $d(\gamma(t_1), \gamma(t_2)) = v|t_1- t_2|$.

It is a known fact that a metric space, in general, may have no geodesics, except constant curves.

I wonder whether there is a well-defined geodesic in the space of compactly supported functions. If there is, how can we compute it? Is the geodesic unique?

I'm specifically interested in the cases of:

- $L^p([0, 1])$ with $L^p$-norm;
- $\mathcal{P}([0, 1])$ with sup-norm, where $\mathcal{P}([0, 1])$ denotes the space of piecewise constant functions over $[0, 1]$.

I'm a newbie to the topic of geometry. So any direct answer or reference to this question would be extremely helpful to me!

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