On open weakly 2-semiconvex sets with nonempty 2-nonsemiconvexity-point sets in three-dimensional Euclidean space

arXiv:2611.14725v1 Announce Type: new Abstract: We study general topological properties of open weakly $2$-semiconvex sets in real three-dimensional Euclidean space $\mathbb{R}^3$. An open set in $\mathbb{R}^3$ is called weakly $2$-semiconvex if through every boundary point of the set there passes a closed half-plane that does not intersect the set. The largest subset of the complement of a given set in $\mathbb{R}^3$ such that every closed half-plane intersecting this subset also intersects the given set is called the $2$-nonsemiconvexity-point set corresponding to the given one. We prove that the nonempty $2$-nonsemiconvexity-point set corresponding to an open weakly $2$-semiconvex set is open and weakly $2$-semiconvex. We also construct a weakly $2$-semiconvex domain whose $2$-nonsemiconvexity-point set is an unbounded domain that can be either convex or nonconvex.

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