Osipchuk T. Analytic and geometric conditions of generalized convex sets

In the thesis properties of generalized linearly convex domains in the hypercomplex spaces which are a direct product of algebras of the real quaternions and Clifford algebras over real numbers field are studied. It is proposed the notions of the left (locally) linearly convex domain and the right one. The necessary and sufficient conditions of the left local linear convexity and the right one of domains of the multi-dimensional quaternion space with smooth boundary are found. For some class of bounded domains with two times smooth boundary of two-dimensional quaternion space, which are the analogs of Hartogs domains the criterions of the left linear convexity and the right one are obtained. Notion of (local) linear convexity is extended to the multi-dimensional Clifford space and the necessary and sufficient conditions of the left local generalized linear convexity and the right one of a domain of this space with two times smooth boundary are obtained.