Geodesic
On three-dimensional bodies of constant width
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 11, Tome 372 (2009), pp. 119-123

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The main results are as follows. Let $K$ be a three-dimensional body of constant width 1, and let $L$ be a line. We denote by $L(K)$ the set of all points where tangent lines of $K$ parallel to $L$ touch $K$. It is proved that for each $L$ the curve $L(K)$ is rectifiable and its length is at most $\sqrt2\pi$; this estimate is sharp. Furthermore, there always exists a line $L$ such that the length of the orthogonal projection of $L(K)$ to $L$ is at most $\sin(\pi/10)+\sin(\pi/20)0.466$. Bibl. – 2 titles. 
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     author = {V. V. Makeev},
     title = {On three-dimensional bodies of constant width},
     journal = {Zapiski Nauchnykh Seminarov POMI},
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     volume = {372},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2009_372_a11/}
}
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V. V. Makeev. On three-dimensional bodies of constant width. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 11, Tome 372 (2009), pp. 119-123. http://geodesic.mathdoc.fr/item/ZNSL_2009_372_a11/