On weakly $1$-convex and weakly $1$-semiconvex sets

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).