# A sharp necessary condition for rectifiable curves in metric spaces

## TL;DR

This paper establishes the most precise necessary condition for a subset of a metric space to be contained in a rectifiable curve, extending classical results and providing sharp criteria in various metric settings.

## Contribution

It proves the sharpest possible converse to Hahlomaa's sufficient condition for doubling curves, advancing the understanding of rectifiability in metric spaces.

## Key findings

- Proves a sharp necessary condition for rectifiable curves in metric spaces.
- Extends results to subsets of metric, Banach spaces, and the Heisenberg group.
- Provides corollaries and implications for rectifiability criteria.

## Abstract

In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane, using a multiscale sum of what is now known as Jones $\beta$-numbers, numbers measuring flatness in a given scale and location. This work was generalized to R^n by Okikiolu, to Hilbert space by the second author, and has many variants in a variety of metric settings. Notably, in 2005, Hahlomaa gave a sufficient condition for a subset of a metric space to be contained in a rectifiable curve. We prove the sharpest possible converse to Hahlomaa's theorem for doubling curves, and then deduce some corollaries for subsets of metric and Banach spaces, as well as the Heisenberg group.}