On transverse hyperplanes to self-similar Jordan arcs

Abstract

We consider self-similar Jordan arcs $\gamma$ in $R^d$, different from a line segment and show that they cannot be projected to a line bijectively. Moreover, we show that the set of points $x\in \gamma$, for which there is a hyperplane, intersecting $\gamma$ at the point x only, is nowhere dense in $\gamma$. .