Prescribed projections and efficient coverings by curves in the plane

Abstract

Davies efficient covering theorem states that an arbitrary measurable set $W$ in the plane can be covered by full lines so that the measure of the union of the lines has the same measure as $W$. This result has an interesting dual formulation in the form of a prescribed projection theorem. In this paper, we formulate each of these results in a nonlinear setting and consider some applications. In particular, given a measurable set $W$ and a curve $\Gamma=\{(t,f(t)): t\in [a,b]\}$, where $f$ is $C^1$ with strictly monotone derivative, we show that $W$ can be covered by translations of $\Gamma$ in such a way that the union of the translated curves has the same measure as $W$. This is achieved by proving an equivalent prescribed generalized projection result, which relies on a Venetian blind construction.