The following is an example that has come up in an application, which is estimating the efficiency of hiking a trail over a given terrain. Consider a two-dimensional vector space, with the positive $x$ direction to be thought of as "uphill." Fix a parameter $i$, which is interpreted as the slope of the hill. In the real application, both $i$ and the uphill direction vary from point to point (as in Riemannian geometry), but for the moment let's say they're constant. For a given vector $\mathbf{v}$, let $\cos\theta$ be the direction cosine with respect to the $x$ axis. Define
$\lVert \mathbf{v} \rVert = \lVert \mathbf{v} \rVert_E f(|i\cos\theta|)$,
where $\lVert \cdot \rVert_E$ is the Euclidean norm and $f$ is a smooth, positive, increasing function with $f(0)=1$. A typical example of an $f$ would be $f(x)=1+x+x^2$, which is the kind of result you get when you measure people's energy consumption on a treadmill.
My $\lVert \cdot \rVert$ has some of the properties of a norm, but it doesn't satisfy the triangle inequality. For a fixed $i$, the possible values of $f$ are bounded above, so this would be a quasinorm, but in the "Riemannian" case where the parameters vary, this may not be true, because $f$ may grow without bound.
In this application, the idea is that someone constructing a trail would build it out of line segments, and they want to minimize the sum of the norms of the segments. The norm is a measure of how much energy it takes a hiker to hike the trail, with the absolute value taken inside $f$ because people will hike the trail in both directions.
I'm interested in properties of quasinorms that would be relevant to this type of example, and e.g. search terms that would lead to more information about quasinorms similar to this one. I've been fiddling around trying to figure out the properties of this quasinorm, and although it's kind of fun, I suspect I may be reinventing the wheel.
One salient property is that geodesics can be nonunique. If $i$ is big enough, and $f$ grows fast enough, then making your way up the hill using switchbacks is more efficient that going straight up it. This violates the triangle inequality. There are two optimal orientations for the switchbacks, but you can make lots of short switchbacks or fewer long ones, and it doesn't affect the length of the path. It seems like you can choose to make these "geodesics" piecewise differentiable, but a typical one would probably be nowhere differentiable.
I would also be interested in measures of how badly a path fails to be a geodesic, and in anything analogous to the calculus of variations and geodesic equations.