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The Intrinsic Core and Minimal Faces of Convex Sets in General Vector Spaces

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The Intrinsic Core and Minimal Faces of Convex Sets in General Vector Spaces The intrinsic core and minimal faces of convex sets in general vector spaces R. D´ıazMill´an∗ and Vera Roshchina† July 19, 2021 Abstract Intrinsic core generalises the finite-dimensional notion of the relative interior to arbitrary (real) vector spaces. Our main goal is to provide a self-contained overview of the key results pertaining to the intrinsic core and to elucidate the relations between intrinsic core and facial structure of convex sets in this general context. We gather several equivalent definitions of the intrinsic core, cover much of the folklore, review relevant recent results and present examples illustrating some of the phenomena specific to the facial structure of infinite-dimensional sets. Keywords: intrinsic core, pseudo-relative interior, inner points, convex sets, general vector spaces, minimal faces MSC2020 Classification: 46N10, 52-02, 52A05. 1 Introduction The relative interior of a convex set in a finite-dimensional real vector space is the interior of this convex set relative to its affine hull; a direct generalisation of this notion to real vector spaces is the intrinsic core [15] (also called pseudo-relative interior [4] and the set of inner points [10]). In contrast to the relative interior in finite dimensions, intrinsic core may be empty for fairly regular sets, which was a key motivation for introducing the notion of quasi-relative interior [5] in the context of topological vector spaces. Even though the quasi-relative interior is a very interesting mathematical object (for instance, [23,24] study duality and separation in the context of quasi-relative interior and resolve a number of open questions), and has much practical importance (e.g. see [3, 6{8, 14, 17{20, 25]), we find that limiting arXiv:2107.07730v1 [math.OC] 16 Jul 2021 our study to the interplay between the intrinsic core and facial structure already offers rich material and provides a valuable perspective on the structure of convex sets in general vector spaces. Such a narrowly focused overview is long overdue: the well-known results and examples discussed here are scattered in the literature, and it may be difficult to find neat references for widely known statements. Besides, we were able to prove some new results and discuss several examples of interesting infinite-dimensional convex sets from the perspective of facial structure (for instance, see Example 2.7, where we construct uncountable chains of faces of the Hilbert cube). ∗Deakin University, Melbourne, Australia †UNSW Sydney, Australia 1 Our take on this topic is consistent with the approach of several recent papers that focus on the intrinsic core. For instance, [9,12,13] study the facial structure of convex sets in linear vector spaces with relation to the intrinsic core, [16] is specifically focused on the intrinsic core of convex cones, and [22] reviews the basic properties of a (less general) notion of the algebraic core, with an emphasis on separation. Notably [16] and [22] do not mention the facial structure that is central to our exposition. We begin with a discussion on faces of convex sets in Section 2, obtaining a new characterisation of minimal faces in terms of the lineality subspace of the cone of feasible directions (Proposition 2.4), leading to a (known) characterisation of a face as an inter- section of line segments (Corollary 2.6). We then talk about chains of faces and present an example of a convex set with uncountable chains of faces (Example 2.7). In Section 3 we discuss several equivalent definitions of intrinsic core, and focus on characterising the intrinsic core in terms of minimal faces: specifically, in Proposition 3.11 we show that the intrinsic core of a convex set coincides with the set of points for which the minimal face is the entire set C, using Proposition 2.4. We also provide an extensive list of properties of the intrinsic core, with self-contained proofs and references. We end the section with the decomposition result (a convex set is a disjoint union of the intrinsic cores of its faces, Theorem 3.14) and a conjecture that every face of a convex set must be the union of a chain of minimal faces (Conjecture 3.17). Intuitively this means that faces of convex sets are always `nested', and never `stacked'. In Section 4 we discuss the notion of algebraic closure and algebraic boundary, and re- late these to the intrinsic core. We also talk about separation and supporting hyperplanes in general real vector spaces, outlining classic separation results pertaining to the setting of general vector spaces, and highlighting the role of intrinsic core in convex separation. We then provide a brief summary in Section 5. 2 Convex sets and their faces Recall that a subset C of a real vector space X is convex if for any two points x; y 2 C we have [x; y] ⊆ C, where [x; y] is the line segment connecting the points x and y, [x; y] = fαx + (1 − α)y j α 2 [0; 1]g: We will also use the notation (x; y) to denote the open line segments connecting x and y, also [x; y) and (x; y] for the relevant half-open segments. Note that algebraically it makes perfect sense to consider degenerate line segments of the form (x; x) = (x; x] = [x; x) = [x; x], since in this case (x; x) = ftx + (1 − t)x j t 2 (0; 1)g = fxg: Even though this notation may appear confusing geometrically, it allows to treat the endpoints of all line segments including the degenerate singleton ones in a unified fashion, and hence streamline much of our discussion. Throughout the paper we assume that X is a real vector space, and that all the sets we consider live in this space (unless stated otherwise). We reiterate this assumption in some of the statements for the ease of reference. A convex subset F of a convex set C is called a face of C if for every x 2 F and every y; z 2 C such that x 2 (y; z), we have y; z 2 F . The set C itself is its own face, and the empty set is a face of any convex set C. A face F of C is proper if F is nonempty and does not coincide with C. We write F C C for the faces F that do not coincide with C and F E C for all faces F of C. Singleton faces that consist of one point only are called 2 extreme points. Some faces of two-dimensional convex sets are shown schematically in Figure 1. Figure 1: Faces of two-dimensional convex sets: proper faces are shown in black. The following two statements are well-known and follow from the definition of a face. Lemma 2.1. Let C be a convex subset of a real vector space X. If F E C and E ⊆ F , then E E F if and only if E E C. Proof. Suppose that C is a convex set, F E C and E ⊆ F . By the definition of a face, E E C is equivalent to having for any x; y 2 C with (x; y) \ E 6= ; that x; y 2 C. However since F is a face of C, and E ⊆ F , any such pair x; y must also be in F (and vice versa, as x; y 2 F ⊂ C). Hence, E E C is equivalent to E E F . Lemma 2.2. Let F be a collection of faces of a convex set C ⊆ X, that is, every member of the set F is a face of C. Then \ E := F F 2F is also a face of C. Proof. First notice that E is a convex subset of C. Now, if for some x; y 2 C the open line segment (x; y) intersects E, then it intersects each F 2 F, and hence [x; y] ⊆ E, so E is a face of C by definition. 2.1 Minimal faces For any subset S of a convex set C there exists a unique minimal face F EC that contains S. We can define minimal faces in a constructive way, with the help of Lemma 2.2. Let S ⊆ C, where C is a convex subset of a real vector space X. The minimal face of C containing S is \ Fmin(S; C) := fF j F E C; S ⊆ F g: The set Fmin(S; C) is a face due to Lemma 2.2, and it is the minimal face (with respect to set inclusion) that contains S. When S = fxg is a singleton we use the notation Fmin(x; C) = Fmin(fxg;C). By cone C we denote the conic hull of C ⊆ X, that is, the set of all finite nonnegative combinations of points in C ⊆ X. ( ) X cone C = αixi j xi 2 C; αi ≥ 0 8i 2 I; jIj < 1 : i2I The conic hull cone C of any set C ⊆ X is a convex cone (it is convex and positively homogeneous, λx 2 K for all x 2 K and λ > 0). When C is convex, we have cone C = R+C = fαx j x 2 C; α ≥ 0g. In particular, when C is convex and x 2 C, then cone(C −x) is the cone of feasible directions of C at x, that is, it consists of the rays along which one can move away from x while staying within C. 3 The lineality space lin K of a nonempty convex cone K ⊆ X is the largest linear subspace contained in K (the lineality space can be defined for general convex sets, but we will only need this notion for cones).
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