A covering theorem for families of sets in $\mbb{R}^{d}$

Let $\mathfrak{D}$ and $\mathfrak{F}$ be families of bounded sets in$\mathbb{R}^{d},$ $d\geq2$. We describe a sense in which $\mathfrak{D}$approximately covers $\mathfrak{F,}$ and we show that then a compact set $X$that covers $\mathfrak{D}$ must also cover $\mathfrak{F}$. As an applicationwe show in the context of Moser’s “worm”problem that a compact set in $\mathbb{R}^{d}$ that contains a congruentcopy of every simple polygonal unit arc is a cover for the family of all unit arcs.

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Published 1 January 2010