TY - JOUR
AU - Cholaquidis, Alejandro
PY - 2023/04/05
Y2 - 2024/08/04
TI - A counter example on a Borsuk conjecture
JF - Applied General Topology
JA - Appl. Gen. Topol.
VL - 24
IS - 1
SE -
DO - 10.4995/agt.2023.18176
UR - https://polipapers.upv.es/index.php/AGT/article/view/18176
SP - 125-128
AB - <p>The study of shape restrictions of subsets of R<sup>d</sup> 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.</p>
ER -