Topological properties of closed weakly m-semiconvex sets

Abstract

The present work considers properties of generally convex sets in the n-dimensional real Euclidean space ℝn, n > 1, known as weakly m-semiconvex, m = 1, 2, … , n − 1. For all that, the subclass of not m-semiconvex sets is distinguished from the class of weakly m-semiconvex sets. A set of the space ℝn is called m-semiconvex if, for any point of the complement of the set to the whole space, there is an m-dimensional half-plane passing through this point and not intersecting the set. An open set of ℝn is called weakly m-semiconvex if, for any point of the boundary of the set, there exists an m-dimensional half-plane passing through this point and not intersecting the given set. A closed set of ℝn is called weakly m-semiconvex if it is approximated from the outside by a family of open weakly m-semiconvex sets. An example of a closed set with three connected components of the subclass of weakly 1-semiconvex but not 1-semiconvex sets in the plane is constructed. It is proved that this number of components is minimal for any closed set of the subclass. An example of a closed set of the subclass with a smooth boundary and four components is constructed. It is proved that this number of components is minimal for any closed, bounded set of the subclass having a smooth boundary and a not 1-semiconvex interior. It is also proved that the interior of a closed, weakly 1-semiconvex set with a finite number of components in the plane is weakly 1-semiconvex. Weakly m-semiconvex but not m-semiconvex domains and closed connected sets in ℝn are constructed for any n ≥ 3 and any m = 1, 2, … , n − 2.