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

We prove that the set of directions of

-dimensional balls which are contained in the boundary

\partial K ∂ K

of a convex body

K \subset {\mathbb R}^n K ⊂ R n

but in no

(n-1) ( n − 1 )

-dimensional convex subset of

\partial K ∂ K

is

\sigma σ

-

1 1

-rectifiable. We also show that there exists a close connection between smallness of the set of directions of line segments on

\partial K ∂ K

and smallness of the set of tangent hyperplanes to the graph of a d. c. (delta-convex) function on

R^{n-2} R n − 2

. Using this connection, we construct

K\subset {\mathbb R}^3 K ⊂ R 3

such that the set of directions of segments on

\partial K ∂ K

cannot be covered by countably many simple Jordan arcs having half-tangents at all points. Also new results on directions of

r r

-dimensional balls in

\partial K ∂ K

parallel to a fixed linear subspace are proved.