Stefanchuk M. Generalized convex sets and its applications

In the thesis there was solved the classical problem of shadow. There was also studied some generalizations of this problem: the problem of shadow for half-convexity, the problem of shadow for arbitrary point of the interior of a sphere, the problem of shadow for the family of sets obtained from a convex set with non-empty interior by a group of geometric transformations which consists from parallel translations and homotheties, the problem of shadow in the complex and hypercomplex spaces. In the n-dimensional real Euclidean space for n>2 there was constructed the set whose n-dimensional lebesgue measure equals zero and this set contains spheres of all radiuses. There was also studied h-hulls of sets and h-extremal elements in the n-dimensional hypercomplex space. Some results concerning multivalued functions in the complex space were generalized into the n-dimensional hypercomplex space: there was proved the hypercomplex analogue of the Fenchel-Moreau theorem and a lot of properties of functions that are conjugate to meaningful functions in the hypercomplex space.