What is the best way to peel fruit?

If the path is allowed to be piecewise smooth (see the comments above), and the fruit is convex, then you can cover the surface with a large number of small patches, and use very short circular trajectories to peel each patch, roughly spinning the blade in place to remove the piece of peel. As the size of patches decreases, this will approach a perfect peel, even if the total length $L$ is chosen to be arbitrarily small. This is like peeling a fruit by bouncing it off a belt sander.

If the fruit is non-convex, we still approach a perfect peeling, as long as we allow $C_K$ to have more than one connected component.

This suggests that the problem is only interesting if you put a bound on the number of jumps.

Sorry, I'm still not allowed to comment. So I use the "Answer" window...

I'm not completely sure to understand your formulation, but for the case of a 2-dimensional sphere and some fixed width of the pealing, you may find your answer in the sphere-filling ropes of Gelrach and von der Mosel. These are ropes with a certain fixed width that are going on a sphere and trying to cover the greatest area. For some width, it is possible to cover everything.

• Heiko von der Mosel et Henryk Gerlach On sphere-filling ropes. Amer. Math. Monthly 118 (2011), no. 10,
• Heiko von der Mosel et Henryk Gerlach : What are the longest ropes on the unit sphere ? Arch. Ration. Mech. Anal. 201 (2011), no. 1, 303–342.