On weakly $(n-1)$-convex sets in low-dimensional Euclidean spaces

Abstract

UDC 514.172

Wе study the properties of generalized convex sets in the real $n$-dimensional Euclidean space $\mathbb{R}^n,$ where $n=2,3,$ known as weakly $(n-1)$-convex sets. An open set in $\mathbb{R}^2$ (respectively, $\mathbb{R}^3$) is called weakly $1$-convex (respectively, weakly $2$-convex) if, for every boundary point of the set, there exists a straight line (respectively, a plane) that passes through this point but does not intersect the indicated set.  A point in the complement of a set in $\mathbb{R}^2$ (respectively, $\mathbb{R}^3$) is called a $1$-nonconvexity (respectively, $2$-nonconvexity) point of the set if every straight line (respectively, every plane) that passes through this point intersects the set. It is shown that each connected component of the $1$-nonconvexity-point set corresponding to a planar weakly $1$-convex set is either the interior of a convex polygon or the interior of a convex generalized polygon. It is proved that the  $2$-nonconvexity-point set corresponding to an open weakly $2$-convex set in $\mathbb{R}^3$ is itself an open and weakly $2$-convex set. We also study some convexity properties of  $2$-nonconvexity-point sets corresponding to weakly $2$-convex sets in $\mathbb{R}^3$.