The topological properties of classes of generally convex sets in multidimensional
real Euclidean space $\mathbb{R}^n$, $n\ge 2$, known as $m$-convex and weakly $m$-convex, $1\le m<n$, are studied in the present work. A set of the space $\mathbb{R}^n$ is called \textbf{\emph{$m$-convex}} if for any point of the complement of the set to the whole space there is an $m$-dimensional plane passing through this point and not intersecting the set. An open set of the space is called \textbf{\emph{weakly $m$-convex}}, if for any point of the boundary of the set there exists an $m$-dimensional plane passing through this point and not intersecting the given set. A closed set of the space is called \textbf{\emph{weakly $m$-convex}} if it is approximated from the outside by a family of open weakly $m$-convex sets. These notions were proposed by Professor Yuri Zelinskii. It is known the topological classification of (weakly) $(n-1)$-convex sets in the space $\mathbb{R}^n$ with smooth boundary. Each such a set is convex, or consists of no more than two unbounded connected components, or is given by the Cartesian product $E^1\times \mathbb{R}^{n-1}$, where $E^1$ is a subset of $\mathbb{R}$. Any open $m$-convex set is obviously weakly $m$-convex. The opposite statement is wrong in general. It is established that there exist open sets in $\mathbb{R}^n$ that are weakly $(n-1)$-convex but not $(n-1)$-convex, and that such sets consist of not less than three connected components. The main results of the work are two theorems. The first of them establishes the fact that for compact weakly $(n-1)$-convex and not $(n-1)$-convex sets in the space $\mathbb{R}^n$, the same lower bound for the number of their connected components is true as in the case of open sets. In particular, the examples of open and closed weakly $(n-1)$-convex and not $(n-1)$-convex sets with three and more connected components are constructed for this purpose. And it is also proved that any compact weakly $m$-convex and not $m$-convex set of the space $\mathbb{R}^n$, $n\ge 2$, $1\le m<n$, can be approximated from the outside by a family of open weakly $m$-convex and not $m$-convex sets with the same number of connected components as the closed set has. The second theorem establishes the existence of weakly $m$-convex and not $m$-convex domains, $1\le m<n-1$, $n\ge 3$, in the spaces $\mathbb{R}^n$. First, examples of weakly $1$-convex and not $1$-convex domains $E^p\subset\mathbb{R}^p$ for any $p\ge3$, are constructed. Then, it is proved that the domain $E^p\times\mathbb{R}^{m-1}\subset\mathbb{R}^n$, $n\ge 3$, $1\le m<n-1$, is weakly $m$-convex and not $m$-convex.
The present work considers properties of classes of generally convex sets in the plane known as 1-semiconvex and weakly 1-semiconvex. More specifically, it is proved that open, weakly 1-semiconvex but not 1-semiconvex set with smooth boundary in the plane consists of not less than four connected components.
We establish a criterion for the local linear convexity of sets in the two-dimensional quaternion space H 2 that are analogs of bounded Hartogs domains with smooth boundary in the two-dimensional complex space C 2 :One of the key notions of many-dimensional complex analysis is the notion of linear convexity. A domain D C n ; n 2; is called locally linearly convex if, for every point z 0 of its boundary @D; there exists a complex hyperplane … C n 1 that passes through the point z 0 and does not intersect the domain D in a certain neighborhood of this point; if, in addition, … C n 1 \ D D ¿; then the domain D is called (globally) linearly convex.The notions of local linear convexity and global linear convexity were introduced by Behnke and Peschl [1], who investigated the domains D D fz D .z 1
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