On transverse hyperplanes to self-similar Jordan arcs

TL;DR

This paper investigates the geometric properties of self-similar Jordan arcs in Euclidean space, demonstrating their non-bijective projection to lines and the sparse nature of points with unique intersecting hyperplanes.

Contribution

It establishes that self-similar Jordan arcs cannot be bijectively projected onto lines and characterizes the distribution of points with unique hyperplane intersections.

Findings

Self-similar Jordan arcs cannot be projected bijectively onto a line.

The set of points with a hyperplane intersecting only at that point is nowhere dense.

Results apply to all self-similar Jordan arcs different from a line segment.

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 $\gamma$. .