# Can we define geodesic in the space of compactly supported functions?

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:

1. $$L^p([0, 1])$$ with $$L^p$$-norm;
2. $$\mathcal{P}([0, 1])$$ 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!

• These are vector spaces, so the geodesic is just a straight line, is it not? That is, the geodesic from $f_1$ to $f_2$ is simply $\gamma(t) = (1-t)f_1 + t f_2$. You can verify this from properties of the norm. Sep 3 '20 at 17:40
• @NateEldridge: for homogenous metrics that is a geodesic, but it is not necessarily unique. Sep 3 '20 at 17:55
• @NateEldredge That's a good catch. I was too involved thinking about the `advanced' geometric ideas... Is there any way to justify the uniqueness? Sep 3 '20 at 18:09
• @mw19930312 I think for $L^p$ it's true and for sup false. At least $l^p$ on $\mathbb{R}^d$ is (Google it), and for sup there's a counterexample on $\mathbb{R}^2$ and you can modify that a bit. Sep 3 '20 at 18:21
• @VilleSalo Thanks for the comments! I'll take a look. Sep 3 '20 at 18:25

A good reference for geodesic spaces (which includes a chapter dedicated to normed vector spaces) is Athanase Papadopoulos, Metric Spaces, Convexity, and Nonpositive Curvature (google books has a preview).

As Nate Eldredge pointed out in the comments, in a normed space you can always form a geodesic between two points, say $$f$$ and $$g$$, as the affine line $$\gamma(t) = (1-t)f + tg$$. However, this geodesic is not always unique. In particular, the $$L^p$$ spaces are uniquely geodesic when $$1, but not for $$p=1$$ or $$p=\infty$$. In fact, any function space with the sup-norm is not uniquely geodesic, as the sup-norm is not strictly convex.