On weakly 1-convex sets in the plane

Abstract

The present work considers the properties of generally convex sets in the plane known as weakly 1-convex. An open set is called weakly 1-convex if for any boundary point of the set there exists a straight line passing through this point and not intersecting the given set. A closed set is called weakly 1-convex if it is approximated from the outside by a family of open weakly 1-convex sets. A point of the complement of a set to the whole plane is called a 1-nonconvexity point of the set if any straight passing through the point intersects the set. It is proved that if an open, weakly 1-convex set has a non-empty set of 1-nonconvexity points, then the latter set is also open. It is also shown that the non-empty interior of a closed, weakly 1-convex set in the plane is weakly 1-convex.