# Sharp estimate on the inner distance in planar domains

## TL;DR

This paper establishes a precise upper bound for the inner distance in planar domains, linking it to the boundary's Hausdorff measure, and introduces new estimates and examples related to Painlevé length.

## Contribution

It provides a sharp estimate for the inner distance in planar domains and advances understanding of Painlevé length bounds with new examples and improved estimates.

## Key findings

- Inner distance is at most the boundary's Hausdorff measure.
- Established an improved Painlevé length estimate for connected sets.
- Provided a totally disconnected example demonstrating the sharpness of the Painlevé length bound.

## Abstract

We show that the inner distance inside a bounded planar domain is at most the one-dimensional Hausdorff measure of the boundary of the domain. We prove this sharp result by establishing an improved Painlev\'e length estimate for connected sets and by using the metric removability of totally disconnected sets, proven by Kalmykov, Kovalev, and Rajala. We also give a totally disconnected example showing that for general sets the Painlev\'e length bound $\kappa(E) \le\pi \mathcal{H}^1(E)$ is sharp.