Topological and geometric properties of the set of 1-nonconvexity points of a weakly 1-convex set in the plane

Abstract

UDC 514.172

In the present work, we consider a class of generalized convex sets in the real plane known as weakly $1$-convex sets.
For a set in the real Euclidean space $\mathbb{R}^n,$ $n\ge 2,$ it is said that a point of the complement of this set to the whole space $\mathbb{R}^n$ is an $\boldsymbol m$-nonconvexity point of the set, $m=\overline{1,n-1},$ if any $m$-dimensional plane passing through this point intersects the set.
An open set in the space $\mathbb{R}^n,$ $n\ge 2,$ is called to be weakly $\boldsymbol m$-convex, $m=\overline{1,n-1},$ if its boundary contains no $m$-nonconvexity points of the set.
Moreover, in the class of open, weakly $1$-convex sets in the plane, we distinguish a subclass of ones with a finite number of connected components and nonempty set of $1$-nonconvexity points.
In this paper, we investigate mainly the properties of the set of $1$-nonconvexity points for the sets from this subclass.
In particular, for any set in this subclass, we prove that the set of its $1$-nonconvexity points is open;
any connected component of the set of its $1$-nonconvexity points is the interior of a convex polygon;
for any convex polygon, there exists a set in this subclass such that its set of $1$-nonconvexity points coincides with the interior of the polygon.