Dakhil H. The shadow problems and mappings of fixed multiplicity

The thesis is devoted to the investigation of geometric and topological properties of generalized convex sets and generalized convex hulls for some families of sets in real, complex and hypercomplex Euclidean space, and the existence of balls mappings of constant mutiplisity. In the thesis there was identified new classes of generalized convex sets and studied their properties, there was solved the problem of shadow for families of balls of a constant radius with freely located centers in three-dimensional Euclidean space, there was found that the problem of shadow for families of balls of a constant radius with centers on a fixed sphere in three-dimensional case doesn't have a finite solution, there was proved that the three balls that do not itersect pairwise always form 1-convex set in three-dimensional Euclidean space, there were built mappings of a fixed multiplicity of interiors of balls which will be homeomorphisms on the boundary.