A counter example on a Borsuk conjecture

Abstract

The  study  of  shape  restrictions  of  subsets  of Rd has  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 ofr-convex supports of absolutely continuous distributions.