On weakly $1$-convex and weakly $1$-semiconvex sets
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917853351903232 |
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| author | Osipchuk, Tetiana M. |
| author_facet | Osipchuk, Tetiana M. |
| contents | The present work concerns generalized convex sets in the real multi-dimensional Euclidean space, known as weakly $1$-convex and weakly $1$-semiconvex sets. An open set is called weakly $1$-convex (weakly $1$-semiconvex) if, through every boundary point of the set, there passes a straight line (a closed ray) not intersecting the set. A closed set is called weakly $1$-convex (weakly $1$-semiconvex) if it is approximated from the outside by a family of open weakly $1$-convex (weakly $1$-semiconvex) sets. A point of the complement of a set to the whole space is a $1$-nonconvexity ($1$-nonsemiconvexity) point of the set if every straight line passing through the point (every ray emanating from the point) intersects the set. It is proved that if the collection of all $1$-nonconvexity ($1$-nonsemiconvexity) points corresponding to an open weakly $1$-convex (weakly $1$-semiconvex) set is non-empty, then it is open. It is also proved that the non-empty interior of a closed weakly $1$-convex (weakly $1$-semiconvex) set in the space is weakly $1$-convex (weakly $1$-semiconvex). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_01022 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On weakly $1$-convex and weakly $1$-semiconvex sets Osipchuk, Tetiana M. General Topology Metric Geometry 52A30 The present work concerns generalized convex sets in the real multi-dimensional Euclidean space, known as weakly $1$-convex and weakly $1$-semiconvex sets. An open set is called weakly $1$-convex (weakly $1$-semiconvex) if, through every boundary point of the set, there passes a straight line (a closed ray) not intersecting the set. A closed set is called weakly $1$-convex (weakly $1$-semiconvex) if it is approximated from the outside by a family of open weakly $1$-convex (weakly $1$-semiconvex) sets. A point of the complement of a set to the whole space is a $1$-nonconvexity ($1$-nonsemiconvexity) point of the set if every straight line passing through the point (every ray emanating from the point) intersects the set. It is proved that if the collection of all $1$-nonconvexity ($1$-nonsemiconvexity) points corresponding to an open weakly $1$-convex (weakly $1$-semiconvex) set is non-empty, then it is open. It is also proved that the non-empty interior of a closed weakly $1$-convex (weakly $1$-semiconvex) set in the space is weakly $1$-convex (weakly $1$-semiconvex). |
| title | On weakly $1$-convex and weakly $1$-semiconvex sets |
| topic | General Topology Metric Geometry 52A30 |
| url | https://arxiv.org/abs/2412.01022 |