Beitraege zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 34 (1993), No. 2, 23-26.
On the Length of Space Curves of Constant Width
Eberhard Teufel
Abstract.
The length of a curve of constant width $d$ in the euclidean plane, i.e.\ the
perimeter of a Gleichdick, is always equal to $\pi d$ (E.~Barbier,
1860). The length of a space curve of constant width $d$ is always greater or
equal to $\pi d$, with equality holding only for plane curves of constant
width
$d$ (H.~B\"uckner, 1937). In this note we obtain an upper
bound for the length of
space curves of constant width (Proposition 1). Furthermore,
we get upper bounds for the length of 1-transnormal curve segments in
non-euclidean planes (see Proposition 2 and Proposition 3).