Simultaneous rotation of infinitely many parallel line segments

TL;DR

This paper extends Davies' 1971 result by demonstrating that a unit square can be rotated so that each vertical line segment sweeps a small area, and applies this to show certain 3D sets have the strong Kakeya property.

Contribution

It proves the unit square can be rotated with each vertical segment sweeping a small area and shows that various 3D sets possess the strong Kakeya property.

Findings

The unit square can be fully rotated with vertical segments sweeping small areas.

Certain 3D sets, like cylindrical surfaces, have the strong Kakeya property.

Abstract

In 1971, Davies proved that finitely many parallel line segments can be simultaneously fully rotated in an arbitrarily small area. In this paper we show that an even stronger statement holds: The unit square can be fully rotated in such a way that each initially vertical line segment sweeps a set of small area. A set in $\mathbb{R}^n$ is said to have the strong Kakeya property if for any two of its positions, the set can be continuously moved between these two positions in an arbitrarily small volume. We use the above result to show that a wide family of sets in $\mathbb{R}^3$, for instance the curved surface of a cylinder, have the strong Kakeya property.