On one-dimensional continua uniformly approximating planar sets

Consider the class of closed connected sets Sigma subset of R-n satisfying length constraint H-1(Sigma) <= l with given l > 0. The paper is concerned with the properties of minimizers of the uniform distance F-M of Sigma to a given compact set M subset of R-n, F-M(Sigma) := max(y is an element of M) dist (y, Sigma), where dist (y, Sigma) stands for the distance between y and Sigma. The paper deals with the planar case n = 2. In this case it is proven that the minimizers (apart trivial cases) cannot contain closed loops. Further, some mild regularity properties as well as structure of minimizers is studied.