Abstract

We prove that the set of directions of (n−2)(n-2)-dimensional balls which are contained in the boundary ∂K\partial K of a convex body K⊂RnK \subset {\mathbb R}^n but in no (n−1)(n-1)-dimensional convex subset of ∂K\partial K is σ\sigma-11-rectifiable. We also show that there exists a close connection between smallness of the set of directions of line segments on ∂K\partial K and smallness of the set of tangent hyperplanes to the graph of a d. c. (delta-convex) function on Rn−2R^{n-2}. Using this connection, we construct K⊂R3K\subset {\mathbb R}^3 such that the set of directions of segments on ∂K\partial K cannot be covered by countably many simple Jordan arcs having half-tangents at all points. Also new results on directions of rr-dimensional balls in ∂K\partial K parallel to a fixed linear subspace are proved.

Author information

Contact details are reproduced from the original publication and may be historical.

David Pavlica

Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic

pavlica@karlin.mff.cuni.cz

Luděk Zajíček

Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic

zajicek@karlin.mff.cuni.cz

Keywords

  • Segments and balls on the boundary of a convex body
  • Hausdorff measure
  • tangent hyperplane
  • d. c. function

Mathematics Subject Classification

52A20; 26B25

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Suggested citation

D. Pavlica, L. Zajíček. “On the Directions of Segments and r-Dimensional Balls on a Convex Surface.” Journal of Convex Analysis 14 (2007), No. 1, 149–167. https://doi.org/10.68381/jca14012

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Published by Heldermann Verlag, 2007. Rights now held by Banach Press LLC.