
Journal of Convex Analysis 14 (2007), No. 1, 149167 Copyright Heldermann Verlag 2007 On the Directions of Segments and rDimensional Balls on a Convex Surface David Pavlica Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic pavlica@karlin.mff.cuni.cz Ludek Zajícek Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic zajicek@karlin.mff.cuni.cz [Abstractpdf] We prove that the set of directions of $(n2)$dimensional balls which are contained in the boundary $\partial K$ of a convex body $K \subset {\mathbb R}^n$ but in no $(n1)$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. (deltaconvex) function on $R^{n2}$. 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 halftangents at all points. Also new results on directions of $r$dimensional balls in $\partial K$ parallel to a fixed linear subspace are proved. Keywords: Segments and balls on the boundary of a convex body, Hausdorff measure, tangent hyperplane, d. c. function. MSC: 52A20; 26B25 [ Fulltextpdf (196 KB)] for subscribers only. 