A counter example on a Borsuk conjecture
Submitted: 2022-08-08
|Accepted:
|Published: 2023-04-05
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Keywords:
r-convex set, locally contractible set, positive reach
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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.
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