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The tube formula is a really nice result in differential geometry which relates the volume of the tubular neighborhood of a submanifold to its intrinsic geometry. It has been proved by Weyl in 1939 who refined some previous results obtained by Hotelling (also in 1939). In the case where the submanifold under study is just a smooth closed curve $\gamma$ in the plane, the result has the following very nice form :

  • If $\epsilon$ is smaller than the least radius of curvature of $\gamma$ the area of the tubular neighborhood of size $\epsilon$ around $\gamma$ is equal to $L\times 2\epsilon$, where $L$ is the length of $\gamma$.

Was this fact about plane curves observed before the late '30s ?

Looking at the proof of this fact, it seems that Darboux or even Gauss would have had all the tools to prove it. Moreover, the question seems to be natural enough to have caught the attention of 19th century geometers.

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