Let $M\subset\Bbb R^n$ be a compact $C^k$ ($k\geqslant 1$) manifold and let $\gamma:(0,1)\to\Bbb R^n$ be a curve of class $C^k$ which is contained in a tubular neighborhood of $M$. Let $\alpha(t)\in M$ be the point minimizing the distance from $\gamma(t)$ to $M$. Then, is $\alpha$ an almost everywhere $C^k$ curve or at least an differentiable almost everywhere curve?
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