Criteria for the Coincidence of the Convex Hulls of the Boundary of a Set and the Closure of this Set

The paper explores the connections between the convex hull of a set, the convex hull of its boundary, and the convex hull of its closure in an affine n-dimensional space An. According to the properties of convex hulls and closures of sets, it follows that the convex hull of a set boundary is contained in the convex hull of the closure of this set. The article examines the conditions for their coincidence. The main result: for a set X in space An, the following statements are equivalent: The convex hull of the boundary of a set coincides with the convex hull of its closure. The closure of the convex hull of the boundary of a set coincides with the closure of the convex hull of the closure of this set. The set X contains no half-spaces. The convex hull of the complement of the set X is the space An. All these propositions are valid for a bounded set. The methods of proof are topological, based on the facts of the theory of convex bodies in An.