A mango made me wonder about this. (See also [this question](https://mathoverflow.net/questions/24054/), 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?



What is the best way to peel fruit?
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.


