What is the best way to peel fruit? A mango made me wonder about this. (See also this question, which is in a similar spirit.)
Fix $L >0$ and a smooth body (possibly nonconvex—pears or bananas are fair game!) $B \subset \mathbb{R}^3$ (and assume w/l/o/g below that $L$ is sufficiently large since we can dilate $B$). For $\gamma:[0,L] \rightarrow \mathbb{R}^3$ smooth and parametrized by arclength and $\theta:[0,L] \rightarrow S^1 $ smooth, let $k(\gamma, \theta,s)$ denote a copy of the unit interval centered at $\gamma(s)$ and in the plane orthogonal to $\dot \gamma(s)$, and at the angle $\theta(s)$ in that plane (we require $k(\gamma,\theta,0)$ to be tangent to $B$, say, and w/l/o/g that this sets $\theta(0) = 0$; angles in planes away from $s=0$ can be sensibly defined via parallel translation). Let $K(\gamma,\theta):= \{ k(\gamma, \theta,s) \cap B : s \in [0,L] \} $. If $K$ contains the boundary of a body $C_K \subset B$ then say that $(\gamma, \theta)$ is a peeling of $B$.
For $L$ fixed, is there an effective way to determine a peeling that minimizes $\mbox{vol}(B \backslash C_K)$? 
Followup: can the best peeling of the unit ball for a given value of $L$ be explicitly constructed?
 A: 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.
A: 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. 

A: After a bit of thought I have a sketch of a very simple case. Suppose that $B$ is a 3-polytope; let $B^*$ denote its dual and consider the graph $G$ associated to the 1-skeleton $B^*_1$ of $B^*$. Now vertices of $G$ correspond to faces of $B$, and edges of $G$ correspond to adjacent faces of $B$. So if $G$ admits a Hamiltonian path then we can use it to get a (nearly?) optimal peeling for $L$ appropriate.
Google results for Hamiltonian circuits on 3-polytopes are here.
