On the Directions of Segments and r-Dimensional Balls on a Convex Surface

We prove that the set of directions of $(n-2)$-dimensional balls which are contained in the boundary $\partial K$ of a convex body $K \subset {\mathbb R}^n$ but in no $(n-1)$-dimensional convex subset of $\partial K$ is $\sigma$-$1$-rectifiable. We also show that there exists a close connection between smallness of the set of directions of line segments on $\partial K$ and smallness of the set of tangent hyperplanes to the graph of a d. c. (delta-convex) function on $R^{n-2}$. Using this connection, we construct $K\subset {\mathbb R}^3$ such that the set of directions of segments on $\partial K$ cannot be covered by countably many simple Jordan arcs having half-tangents at all points. Also new results on directions of $r$-dimensional balls in $\partial K$ parallel to a fixed linear subspace are proved.