Topological and geometric properties of 1-nonsemiconvexity-point sets corresponding to open weakly 1-semiconvex subsets of the plane

The present work concerns generalized convex sets in the Euclidean plane, known as weakly 1-semiconvex. An open set is called weakly 1-semiconvex if every boundary point of the set is the initial point of a closed ray not intersecting the set. A closed set is called weakly 1-semiconvex if it can be approximated from the outside by a family of open weakly 1-semiconvex sets. A point of the complement of a set to the whole plane is a 1-nonsemiconvexity point of the set if every closed ray emanating from the point intersects the set. It is proved that if the 1-nonsemiconvexity-point set corresponding to an open weakly 1-semiconvex set is non-empty, then it is also open and weakly 1-semiconvex, and its arbitrary connected component is a bounded convex set such that any connected part of its boundary consisting of only smooth points is a line segment or a point. It is also proved that the non-empty interior of a closed weakly 1-semiconvex set in the plane is weakly 1-semiconvex.