Asphericity of shadows of a convex body

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.