A chord-arc covering theorem in Hilbert space

TL;DR

This paper proves a covering theorem in Hilbert space, showing that any rectifiable curve can be covered by a finite union of chord-arc curves with controlled total length, enhancing understanding of geometric structures in infinite-dimensional spaces.

Contribution

It introduces a new covering theorem for rectifiable curves in Hilbert space, establishing bounds on coverings by chord-arc curves with finite total length.

Findings

Almost every point on the curve is contained in a countable union of chord-arc curves.

The total length of these chord-arc curves is bounded by a constant times the original curve's length.

The theorem extends geometric measure theory concepts to infinite-dimensional Hilbert spaces.

Abstract

We prove that there exists $M>0$ such that for any closed rectifiable curve $\Gamma$ in Hilbert space, almost every point in $\Gamma$ is contained in a countable union of $M$ chord-arc curves whose total length is no more than $M$ times the length of $\Gamma$.