The present work considers properties of generally convex sets in real three-dimensional space R3 known as weakly 2-convex. An open set in R3 is called weakly 2-convex if through every boundary point of the set there passes a plane not intersecting the given set. A closed set in R3 is called weakly 2-convex if it is approximated from the outside by a family of open weakly 2-convex sets. A point of the complement of a set in R3 to the whole space is a 2-nonconvexity point of the set if every plane passing through the point intersects the set. It is proved that the non-empty interior of a closed weakly 2-convex set is weakly 2-convex. It is shown that for any convex polyhedron there exists an open weakly 2-convex set such that its 2-nonconvexity-point set coincides with the polyhedron interior, but there exists an open weakly 2-convex set such that its open bounded convex 2-nonconvexity-point set differs from the interior of a convex polyhedron.
In this paper, lower bound of exponential type for the distortion of the transfinite diameter of the image of a disk under regular homeomorphisms belonging to the Sobolev class $W_{\rm loc }^{1,1}$ and possessing the N-property (in the sense of Luzin) is obtained. Moreover, the problem of minimizing the distortion functional of the transfinite diameter of the image of a disk in a certain class of such mappings is solved.
In this work, special pseudo-Riemannian spaces endowed with an affinor structure satisfying certain naturally arising conditions are investigated. Spaces with a special form of the Riemann tensor are considered, and two types of pseudo-Riemannian spaces are distinguished depending on the type of their Riemann tensor. Their geometric characteristics are obtained for a specific form of the metric tensor. In particular, it is proven that pseudo-Riemannian spaces of constant almost holomorphic curvature with an elliptic skew-symmetric affinor structure of second degree are semi-symmetric. The research is carried out locally, using tensor methods without imposing restrictions on the signature or definiteness of the metric tensor of the space under consideration. A special conjugation operation, introduced for spaces endowed with an affinor structure, is widely applied, along with its properties for the Riemann and Ricci tensors.
The topological properties of flows with Hamiltonian dynamics on the Boy's surface, considered as a stratified set, are investigated. Here, 1-strata are simple trajectories of the flow, and planar angles in the neighborhood of the 0-stratum consist of one or two hyperbolic sectors. On each stratum, the Hamiltonian of the flow is a simple Morse function. A topological classification of such simple flows is constructed using distinguishing graphs consisting of rooted trees (Reeb graphs). A recurrence formula is obtained for calculating the number of topologically non-equivalent flows with a given number of separatrices.
This paper investigates the class of mappings that satisfy the upper weighted Poletsky inequality. We focus on the setting where the mapping is not assumed to be boundary-preserving, and the corresponding majorant satisfies the integral-type restrictions. By imposing specific geometric constraints on the definition domain and respective cluster sets of mappings, we establish the equicontinuity of the corresponding families of mappings in the closure of a domain.
