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

TL;DR

This paper investigates generalized convex sets called weakly 1-convex and weakly 1-semiconvex sets, establishing properties of their nonconvexity points and the structure of their interiors in Euclidean space.

Contribution

It introduces and analyzes the properties of weakly 1-convex and weakly 1-semiconvex sets, including the openness of nonconvexity points and the convexity of their interior.

Findings

Non-empty nonconvexity points form an open set.

Interior of a closed weakly 1-convex set is weakly 1-convex.

Properties extend to both open and closed weakly 1-convex 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).