Topological and geometric properties of the set of 1-nonconvexity points of a weakly 1-convex set in the plane
DOI:
https://doi.org/10.37863/umzh.v73i12.6890Keywords:
convex set, m-convex set, weakly m-convex set, set of m-nonconvexity points, Euclidean spaceAbstract
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 Rn, n≥2, it is said that a point of the complement of this set to the whole space Rn 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.
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