On weakly 1-convex and weakly 1-semiconvex sets in real Euclidean spaces

Abstract

The present work concerns generalized convex sets in the real multi-dimensional Euclidean space, known as weakly 1-convex and weakly 1-semiconvex sets. An open set is called weakly 1-convex (weakly 1-semiconvex) if, through every boundary point of the set, there passes a straight line (a closed ray) not intersecting the set. A closed set is called weakly 1-convex (weakly 1-semiconvex) if it is approximated from the outside by a family of open weakly 1-convex (weakly 1-semiconvex) sets. A point of the complement of a set to the whole space is a 1-nonconvexity (1-nonsemiconvexity) point of the set if every straight line passing through the point (every ray emanating from the point) intersects the set. It is proved that if the collection of all 1-nonconvexity (1-nonsemiconvexity) points corresponding to an open weakly 1-convex (weakly 1-semiconvex) set is non-empty, then it is open. It is also proved that the non-empty interior of a closed weakly 1-convex (weakly 1-semiconvex) set in the space is weakly 1-convex (weakly 1-semiconvex).