On three-dimensional bodies of constant width

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.