Vyhovska I. Combinatorial theorems and convexity criteria for generalized convex sets

We study combinatorial theorems and criteria of convexity and generalized convexity for subsets of Euclidean space. We construct a chain of iterated shells set, which is an intermediate stage in the construction of convex hulls. We establish new external and boundary criteria for convexity domains and compact sets in Euclidean space. For acyclic compacts found boundary criterion of strong linear convexity. The class of generalized convex sets with a dense set of points of smoothness on the boundary is introduced, which well approximate strongly linear convex sets with an arbitrary boundary. Criterion for strong linear convexity Cartesian product of compacts in multidimensional hypercomplex space is established.