On the Directions of Segments and r-Dimensional Balls on a Convex Surface
Abstract
<jats:p> We prove that the set of directions of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">(n-2)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -dimensional balls which are contained in the boundary <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\partial K</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> of a convex body <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">K \subset {\mathbb R}^n</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> but in no <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">(n-1)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -dimensional convex subset of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\partial K</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\sigma</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>σ</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> - <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">1</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -rectifiable. We also show that there exists a close connection between smallness of the set of directions of line segments on <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\partial K</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> and smallness of the set of tangent hyperplanes to the graph of a d. c. (delta-convex) function on <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">R^{n-2}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>R</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . Using this connection, we construct <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">K\subset {\mathbb R}^3</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> such that the set of directions of segments on <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\partial K</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> cannot be covered by countably many simple Jordan arcs having half-tangents at all points. Also new results on directions of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">r</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -dimensional balls in <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\partial K</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> parallel to a fixed linear subspace are proved. </jats:p>