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4 inserted parallel translation blurb

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 $\theta(0) = 0$)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$be explicitly constructed? 3 added 44 characters in body A mango made me wonder about this. (See also this question, which is in a similar spirit.) Fix$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$, say)say, and w/l/o/g that this sets$\theta(0) = 0$). 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$be explicitly constructed? 2 deleted 26 characters in body A mango made me wonder about this. (See also this question, which is in a similar spirit.) Fix$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)$and in the parallel transport of the plane orthogonal to$\dot \gamma(s)$, and at the angle$\theta(s)$in that plane (we require$k(\gamma,\theta,0)$to be tangent to$B$, say). 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?

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