Intersection of continua and rectifiable curves

TL;DR

The paper proves that every non-degenerate continuum in Euclidean space intersects with some rectifiable curve in a set of Hausdorff dimension 1, answering a longstanding question in geometric measure theory.

Contribution

It establishes the existence of rectifiable curves intersecting continua with Hausdorff dimension 1, advancing understanding of geometric intersections in Euclidean spaces.

Findings

Existence of rectifiable curves intersecting continua with Hausdorff dimension 1

Answers a question posed by B. Kirchheim

Contributes to geometric measure theory

Abstract

We prove that for any non-degenerate continuum $K \subseteq \mathbb{R}^d$ there exists a rectifiable curve such that its intersection with $K$ has Hausdorff dimension 1. This answers a question of B. Kirchheim.