Simultaneous rotation of infinitely many parallel line segments

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 dollar sign double struck upper R n dollar sign ${\mathbb{R}}^n$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>ℝ</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> 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 dollar sign double struck upper R 3 dollar sign ${\mathbb{R}}^3$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>ℝ</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> </mml:math> , for instance, the lateral surface of a cylinder, have the strong Kakeya property.