On weakly <i>m</i>-convex sets

Dakhil, H.K.; Zelinskii, Yu. B.; Klishchuk, B. A.

doi:10.15407/dopovidi2017.04.003

Cited by 2 publications

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“…The geometric and topological properties of weakly m-convex sets are investigated in [3]. In particular, the following proposition is proved in [3]:…”

Section: Definition 14 ([1]) a Set A Is Approximated From The Outside...mentioning

confidence: 99%

“…We will prove the theorem by presenting examples of appropriate sets. First, construct domains in the space R 3 of the class WC 3 1 zC 3 1 approximating from the outside a closed connected set belonging to the same class. Consider the open sets…”

Section: Theorem 33mentioning

confidence: 99%

“…However, E 3 0 is not 1-convex due to Lemma 3.1. Hence, the closed and connected set E 3 0 belongs to the class WC 3 1 zC 3 1 .…”

Section: Ifmentioning

confidence: 99%

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On closed weakly m-convexsets

Осіпчук¹

2022

PIGC

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In the present work we study properties of generally convex sets in the n-dimensional real Euclidean space Rn, (n>1), known as weakly m-convex, m=1,...,n-1. An open set of Rn is called weakly m-convex if, for any boundary point of the set, there exists an m-dimensional plane passing through this point and not intersecting the given set. A closed set of Rn is called weakly m-convex if it is approximated from the outside by a family of open weakly m-convex sets. A point of the complement of a set of Rn to the whole space is called an m-nonconvexity point of the set if any m-dimensional plane passing through the point intersects the set. It is proved that any closed, weakly (n-1)-convex set in Rn with non-empty set of (n-1)-nonconvexity points consists of not less than three connected components. It is also proved that the interior of a closed, weakly 1-convex set with a finite number of components in the plane is weakly 1-convex. Weakly m-convex domains and closed connected sets in Rn with non-empty set of m-nonconvexity points are constructed for any n>2 and any m=1,...,n-1.

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“…The geometric and topological properties of weakly m-convex sets are investigated in [3]. In particular, the following proposition is proved in [3]:…”

Section: Definition 14 ([1]) a Set A Is Approximated From The Outside...mentioning

confidence: 99%

Section: Theorem 33mentioning

confidence: 99%