On the Convex Hull of the Boundary of a Set
УДК 514.172
Abstract
The article studies the relations between the convex hulls of the boundary of a set and the closure of this set in n-dimensional Affine space An. In the previous work, the authors found criteria for the coincidence of the convex hull of the boundary of a set with the convex hull of its closure. This paper presents a description of the closures of the convex hulls of the boundary of a set and the closure of this set in the case where there is no coincidence.
Main result.
Theorem. If the convex hull of the boundary of a set X in An does not coincide with the convex hull of the closure of the set X, then one of two cases occurs:
- The convex hull of the closure of a set X is a space An.
- The convex hull of the closure of a set X is a closed half-space, and the closure of the convex hull of the boundary of the set X is either a hyperplane bounding this half-space, or a layer between two parallel hyperplanes (including these hyperplanes), one of which bounds this half-space.
Methods of proof — topological, based on the theory of convex sets in An.
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References
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Copyright (c) 2025 Ирина Викторовна Поликанова, Мария Викторовна Куркина
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