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Internal structure of convex sets and their faces

V. V. Gorokhovik

Institute of Mathematics of the National Academy of Sciences of Belarus, Minsk

Abstract: Most often, the geometric structure of convex sets is associated with their facial structure. In the first section of this paper, we present a somewhat different approach to characterizing the geometric structure of convex sets based on the concept of an open component of a convex set. In this paper, we consider convex sets in infinite-dimensional real vector spaces endowed with no topology. To define the notion of an open component of a convex set $Q$, the preorder relation $\unlhd_Q$ is introduced on $Q$ (its own for each set $Q$) called a dominance relation. Open components of a convex set $Q$ are defined as equivalence classes of the quotient set $Q/\mathbin{<\!>}_Q$ of the set $Q$ by the equivalence relation $\mathbin{<\!>}_Q$, which is the symmetric part of the dominance relation $\unlhd_Q$. Each open component of a convex set $Q$ is a relatively algebraic open subset of the set $Q$ under consideration, and the set $Q$ is a disjoint union of all open components belonging to $Q$. The dominance relation $\unlhd_Q$ induces a partial order relation $\unlhd_Q^*$ on the family ${\mathcal O}(Q):= Q/\mathbin{<\!>}_Q$ of all open components of the set $Q$ with respect to which the partially ordered family $({\mathcal O}(Q),\unlhd_Q^*)$ is an upper semilattice. For halfspaces (convex sets $H$ whose complements are also convex), the corresponding upper semilattice $({\mathcal O}(H),\unlhd_H^*)$ is a linearly ordered set. The internal structure of a convex set $Q$ is identified in the paper with the structure of the upper semilattice $({\mathcal O}(Q),\unlhd_Q^*)$. In the second section of the paper, the connection between the internal structure of a convex set and that of its faces is investigated. It is established that each open component of a convex set $Q$ is a relative algebraic interior of the minimal (with respect to inclusion) face of $Q$ containing the given open component. Conversely, if a face $F$ of a convex set $Q$ has a nonempty relative algebraic interior, then it (the relative algebraic interior of the face) coincides with some open component of the set $Q$, and the face $F$ itself is a minimal face containing this open component (such faces are called minimal in the paper). In finite-dimensional vector spaces, any face $F$ of a convex set $Q$ is minimal, whereas in any infinite-dimensional vector space, there exist convex sets whose faces are not all minimal. Concurrently, each open component of any face $F$ of a convex set $Q$ is an open component of $Q$ itself; i.e., ${\mathcal O}(F) \subset {\mathcal O}(Q)$. Moreover, the partial order relation $\unlhd_F^*$ defined on ${\mathcal O}(F)$ coincides with the restriction to ${\mathcal O}(F)$ of the partial order relation $\unlhd_Q^*$ defined on ${\mathcal O}(Q)$. Thus, the internal structure $({\mathcal O}(F),\unlhd_F^*)$ of any face $F$ of a convex set $Q$ is a substructure of the internal structure $({\mathcal O}(Q),\unlhd_Q^*)$ of $Q$ itself.

Keywords: convex sets, halfspaces, faces, open component, semilattice, preorder, linear order.

UDC: 514.172

MSC: 52A05, 52A99

Received: 14.02.2025
Revised: 18.03.2025
Accepted: 24.03.2025

DOI: 10.21538/0134-4889-2025-31-2-fon-04

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