A covering theorem for families of sets in Rd

More than 40 years ago Leo Moser [7] (reprinted in W. Moser [8]) asked for the set of least area in the plane that can cover every planar arc of unit length. A great variety of similar problems can be formulated by restricting the covering set to be convex, or of prescribed shape (e.g., triangular, etc.), by seeking to minimize the perimeter, or thickness, or some other geometric measure, or by restricting the family of unit arcs in various ways (to be closed, or polygonal, or polygonal with at most n segments, etc.). Very few of these problems, known collectively as “worm” problems, have been solved, and the analogous problems in higher dimensions are virtually unexplored. For a recent survey of the status of Moser’s original problem in the plane, see Wetzel [11], and for a glimpse at the closely related notion of “escape” path, see Finch and Wetzel [4]. In this article we establish a covering theorem for families of bounded sets in Rd, and we apply it to prove a covering approximation lemma that is potentially useful in “worm” problems in higher dimensions.