An $r$-convex set which is not locally contractible
Abstract
The study of shape restrictions of subsets of have several applications in many areas, being convexity, -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 -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 -dimensional Lebesgue measure), the problem is much more difficult. We present a counter example of a not-locally contractible set, which is -convex. This also proves that the class of supports with positive reach of absolutely continuous distributions includes strictly the class of -convex supports.
Keywords
Cite
@article{arxiv.2211.08107,
title = {An $r$-convex set which is not locally contractible},
author = {Alejandro Cholaquidis},
journal= {arXiv preprint arXiv:2211.08107},
year = {2022}
}