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, we say that a point of the complement of this set to the entire space ℝn is an m-nonconvexity point of the set, \( m=\overline{1,n-1} \), if any m-dimensional plane passing through this point crosses the indicated set. An open set in the space ℝn, n ≥ 2, is called weakly m-convex, \( m=\overline{1,n-1} \), if its boundary does not contain any m-nonconvexity points of the set. Moreover, in the class of open weakly 1-convex sets in the plane, we select a subclass of sets with finitely many connected components and a nonempty set of 1-nonconvexity points. We mainly analyze the properties of the set of 1-nonconvexity points for the sets from the indicated subclass. In particular, for any set in this subclass, it is proved that the set of its 1-nonconvexity points is open, that any connected component of the set of its 1-nonconvexity points is the interior of a convex polygon, and that, for any convex polygon, there exists a set from the indicated subclass such that its set of 1-nonconvexity points coincides with the interior of a polygon.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, No. 12, pp. 1657–1672, December, 2021. Ukrainian DOI: https://doi.org/10.37863/umzh.v73i12.6890.
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Osipchuk, T.M. Topological and Geometric Properties of the Set of 1-Nonconvexity Points of a Weakly 1-Convex Set in the Plane. Ukr Math J 73, 1918–1936 (2022). https://doi.org/10.1007/s11253-022-02038-w
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DOI: https://doi.org/10.1007/s11253-022-02038-w