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 $\mathbb{R}^{n}$, $n>1$, known as weakly $m $-semiconvex, $m=1,2,\ldots ,n-1$. 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 $\mathbb{R}^{n}$ is called \textbf{\emph{$m$-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 $\mathbb{R}^{n}$ 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 $\mathbb{R}^{n}$ is called \textbf{\emph{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 $\mathbb{R}^{n}$ are constructed for any $n\geq 3$ and any $m=1,2,\ldots ,n-2$.