# Wetzel's sector covers unit arcs

## TL;DR

This paper proves that a 30-degree circular sector of unit radius can contain any planar arc of the same length, settling a long-standing conjecture and identifying the smallest known convex region for this purpose.

## Contribution

The paper confirms Wetzel's 1970s conjecture and establishes the 30-degree sector as the minimal convex region capable of accommodating all unit-length planar arcs.

## Key findings

- A 30° sector of radius 1 can contain all unit-length planar arcs.
- This sector is the smallest known convex region with this property.
- The result settles a 50-year-old conjecture in geometric covering problems.

## Abstract

We settle J. Wetzel's 1970's conjecture and show that a 30{^\circ} circular sector of unit radius can accommodate every planar arc of unit length. Leo Moser asked in 1966 for the smallest (convex) region in the plane that can accommodate each arc of unit length. With area {\pi}/12, this sector is the smallest such set presently known. Moser's question has prompted a multitude of papers on related problems over the past 50 years, most remaining unanswered.