Geodesic
Asphericity of shadows of a convex body
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 9, Tome 329 (2005), pp. 67-78 
V. V. Makeev. Asphericity of shadows of a convex body. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 9, Tome 329 (2005), pp. 67-78. http://geodesic.mathdoc.fr/item/ZNSL_2005_329_a5/
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     title = {Asphericity of shadows of a~convex body},
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

A shadow is a parallel projection $F$ of a body $K$ to a plane. $F$ is $\epsilon$-aspheric if the boundary $\partial F$ lies in a circular ring with center at $O$ and ratio of radii equal to $1+\epsilon$. $F$ is $\epsilon$-aspheric for a part of $\alpha$ if the same is true for the part of $\partial F$ lying inside an angle of $2\alpha\pi$ with vertex at $O$ (or within the union of two vertical angles of $\alpha\pi$ if $K$ is centrally symmetric). It is proved that each convex body $K\subset\mathbb R^3$ has a $(\sqrt 2-1)$-aspheric shadow and a shadow $(\sec\pi/5-1)$-aspheric for a part of 4/5. If $K$ is centrally symmetric, then $K$ has a $(2/\sqrt3-1)$-aspheric shadow and a shadow $(\sec\pi/7-1)$-aspheric for a part of 6/7.

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