Minimal length curves that are not embeddable in an open planar set: the problem of a lost swimmer with a compass

Given an open bounded set S in R², the problem of computing a path f of minimum size such that for every x member of S the set {x}+f intersects the boundary of S is considered. The existence of such paths is proved both when the path size is its length and when it is its (one-dimensional Hausdorff outer) measure. Some theorems characterizing optimal paths are proved and it is shown that when S is convex, the minimum width chords of Cl(S) are optimal with respect to both size definitions.