An $r$-convex set which is not locally contractible

Abstract

The study of shape restrictions of subsets of $\mathbb{R}^d$ have several applications in many areas, being convexity, $r$-convexity, and positive reach, some of the most famous, and typically imposed in set estimation. The following problem was attributed to K. Borsuk, by J. Perkal in 1956: find an $r$-convex set which is not locally contractible. Stated in that way is trivial to find such a set. However, if we ask the set to be equal to the closure of its interior (a condition fulfilled for instance if the set is the support of a probability distribution absolutely continuous with respect to the $d$-dimensional Lebesgue measure), the problem is much more difficult. We present a counter example of a not-locally contractible set, which is $r$-convex. This also proves that the class of supports with positive reach of absolutely continuous distributions includes strictly the class of $r$-convex supports.