� EXTRACTA MATHEMATICAE Volumen 33, Número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura EXTRACTA MATHEMATICAE Vol. 36, Num. 2 (2021), 241 – 278 doi:10.17398/2605-5686.36.2.241 Available online October 21, 2021 Support and separation properties of convex sets in finite dimension V. Soltan Department of Mathematical Sciences, George Mason University 4400 University Drive, Fairfax, VA 22030, USA vsoltan@gmu.edu Received August 21, 2021 Presented by H. Martini Accepted September 20, 2021 Abstract: This is a survey on support and separation properties of convex sets in the n-dimensional Euclidean space. It contains a detailed account of existing results, given either chronologically or in related groups, and exhibits them in a uniform way, including terminology and notation. We first discuss classical Minkowski’s theorems on support and separation of convex bodies, and next describe various generalizations of these results to the case of arbitrary convex sets, which concern bounding and asymptotic hyperplanes, and various types of separation by hyperplanes, slabs, and complementary convex sets. Key words: Convex, cone, bound, hyperplane, support, separation. MSC (2020): 52A20. Contents 1 Introduction 242 2 Preliminaries 243 3 Minkowski’s theorems 247 3.1 Support hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . 247 3.2 Bounding and separating hyperplanes . . . . . . . . . . . . . . 250 3.3 Sufficient conditions for convexity of solid sets . . . . . . . . . . 252 4 Supports and bounds of convex sets 254 4.1 Support hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . 254 4.2 Bounding hyperplanes and halfspaces . . . . . . . . . . . . . . . 256 5 Separation of convex sets 260 5.1 Classification of separating hyperplanes . . . . . . . . . . . . . 260 ISSN: 0213-8743 (print), 2605-5686 (online) © The author(s) - Released under a Creative Commons Attribution License (CC BY-NC 3.0) https://doi.org/10.17398/2605-5686.36.2.241 mailto:vsoltan@gmu.edu https://publicaciones.unex.es/index.php/EM https://creativecommons.org/licenses/by-nc/3.0/ 242 v. soltan 5.2 Geometric conditions on hyperplane separation . . . . . . . . . 264 5.3 Sharp separation of convex cones . . . . . . . . . . . . . . . . . 267 5.4 Penumbras and separation . . . . . . . . . . . . . . . . . . . . . 269 5.5 Hemispaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 1. Introduction Support and separation properties of convex sets are among the core topics of convexity theory. Introduced and studied by the prominent mathematician Hermann Minkowski (see [63] and [64]), they became useful tools of convex geometry. Further development of Minkowski’s ideas occurred at the begin- ning of 20th century and was summarized in the monograph of Bonnesen and Fenchel [8]. A rapid growth of linear analysis in the first half of 20th century led to numerous generalizations of Minkowski’s contribution to the case of infinite- dimensional vector spaces and made this topic an organic part of the discipline. For instance, the survey of Klee [51] from 1969 describes existing results on support and separation properties of convex sets in vector spaces of any di- mension, with sporadic divisions into finite- and infinite-dimensional cases. However, besides obvious similarities between these cases, they display nowa- days different goals: while finite-dimensional convexity deals, predominantly, with the properties of convex sets, a big part of similar results in linear analysis became a tool for various classifications of topological and normed spaces. A new wave of interest towards finite-dimensional convexity occurred in the second half of the 20th century due to advancements in linear program- ming, with polyhedral sets considered as geometric interpretation of solutions sets of systems of linear inequalities (see, e.g., Dantzig [19] and Černikov [18]), and later in convex analysis, optimization theory, and polyhedral geometry. Various books on convex analysis (see, e.g., Güler [37], Panik [69], and Rock- afellar [72]) contain separate chapters on support and separation properties of convex sets, illustrating a continuous development of these topics. These books and numerous articles in the field also underline a shift of interest from the study of convex bodies toward arbitrary convex sets, possibly unbounded or nonclosed. Despite a steady progress in research, no comprehensive survey on support and separation properties of convex sets in finite dimension was published dur- ing the last five decades. The present paper aims to fill in (at least partly) this support and separation properties 243 gap and overview new trends and results. It contains an account of existing facts, given either chronologically or in related groups, exhibiting them in a uniform way, including terminology and notation. We do not consider here al- gorithmic and computational aspects of the theory on support and separation of convex sets, which deserve their own surveys. Following the necessary preliminaries, given in Section 2, the main text is divided into three parts. Section 3 describes classical Minkowski’s theorems on support and separation properties of convex bodies and summarizes immediate contributions of his colleagues. Contemporary approach to the study of these topics is given in Sections 4 and 5. These sections cover existing results on support, bounding and asymptotic planes of arbitrary convex sets, and various types of separation of convex sets by hyperplanes, slabs, and complementary convex halfspaces. It is interesting to compare methods of research on support and separation of convex sets in classical and contemporary periods. Since the main results of the classical period are related to full-dimensional compact sets, the methods of their proofs are predominantly based on compactness arguments and basic topology of open sets in finite dimension. Contemporary results in this field deal with arbitrary convex sets, possibly unbounded and having intermediate dimension. Consequently, their proofs employ the concept of relative interior and extensively use various types of cones associated with convex sets. 2. Preliminaries This section describes necessary notation, terminology, and results on con- vex sets in the n-dimensional Euclidean space Rn (see, e.g., [76] for details). The elements of Rn are called vectors (or points), and o stands for the zero vector of Rn. An r-dimensional plane L in Rn, where 0 ≤ r ≤ n, is a translate, L = a + S, a ∈ Rn, of a suitable r-dimensional subspace S of Rn, called the direction space of L. A hyperplane is a plane of dimension n − 1; it can be described as H = {x ∈ Rn : x·e = γ}, e 6= o, γ ∈ R, (2.1) where x·e means the dot product of vectors x and e. Nonzero multiples λe of the vector e in (2.1) are called normal vectors of H. The direction space of the hyperplane (2.1) is the (n− 1)-dimensional subspace given by S = {x ∈ Rn : x·e = 0}, e 6= o. (2.2) 244 v. soltan Every hyperplane of the form (2.1) determines the opposite closed halfs- paces V1 = {x ∈ Rn : x·e ≤ γ} and V2 = {x ∈ Rn : x·e ≥ γ} and the pair of opposite open halfspaces W1 = {x ∈ Rn : x·e < γ} and W2 = {x ∈ Rn : x·e > γ}. The closed and open (line) segments with distinct endpoints u and v in Rn and the halfline through v with endpoint u are denoted by [u,v], (u,v), and [u,v〉, respectively. The norm (or the length) of a vector x ∈ Rn is denoted ‖x‖. Given a point a ∈ Rn and a scalar ρ > 0, the sphere and balls (closed and open) of radius ρ and center a are denoted Sρ(a), Bρ(a), and Uρ(a), respectively. The topological interior, closure, and boundary of a nonempty set X ⊂ Rn are given by int X, cl X, and bd X, respectively. The open ρ-neighborhood of X, denoted Uρ(X), is the union of all open balls Uρ(x) of radius ρ > 0 centered at x ∈ X. Nonempty sets X1 and X2 in Rn are called strongly disjoint provided Uρ(X1)∩Uρ(X2) = ∅ for a suitable ρ > 0; the latter occurs if and only if the inf -distance δ(X1,X2), defined by δ(X1,X2) = inf{‖x1 −x2‖ : x1 ∈ X1,x2 ∈ X2}, is positive. For a nonempty set X ⊂ Rn, the notations span X and X⊥, stand, respectively, for the span and orthogonal complement of X. The set X is called proper if ∅ 6= X 6= Rn. The affine span of X, denoted aff X, is the intersection of all planes con- taining X, and dim X is defined as the dimension of the plane aff X. Also, the direction space of X is defined by dir X = aff X − aff X, and the orthogonal space of X by ort X = (dir X)⊥ (generally, ort X 6= X⊥). Given nonempty sets X,Y ⊂ Rn and a scalar λ, we let X + Y = {x + y : x ∈ X,y ∈ Y}, λX = {λx : x ∈ X}. Nonempty sets X and Y in Rn are called (directly) homothetic provided X = z + λY for suitable z ∈ Rn and λ > 0. In what follows, K means a convex set in Rn. To avoid trivial cases, we will be assuming that all convex sets involved are nonempty. A point x of a convex set K ⊂ Rn is said to be relatively interior to K provided there is a scalar ρ = ρ(x) > 0 such that aff K ∩ Uρ(x) ⊂ K. The set of all relatively support and separation properties 245 interior points of K is called the relative interior of K and is denoted rint K. The set rint K is nonempty and convex; furthermore, cl K = cl (rint K). The difference between topological and relative interiors can be illustrated by the following example: If K is a unit circular disk of the coordinate xy-plane of R3, then int K = ∅, while rint K is the interior of this disk. The relative boundary of a convex set K ⊂ Rn, denoted rbd K, is defined by rbd K = cl K \ rint K. It is known that rbd K 6= ∅ if and only if K is not a plane. A convex body in Rn is a compact convex set with nonempty interior, a (convex) polyhedron is the intersection of finitely many closed halfspaces, and a polytope (bounded polyhedron) is the convex hull of finitely many points. A contemporary approach to the study of support and separation prop- erties of convex sets deals with a family of various associated cones. We recall that a nonempty set C in Rn is called a cone with apex a ∈ Rn if a + λ(x − a) ∈ C whenever λ ≥ 0 and x ∈ C. (Obviously, this definition implies that a ∈ C, although a stronger condition λ > 0 can be beneficial; see, e.g., [57].) The cone C is called convex if it is a convex set. The apex set of a convex cone C ⊂ Rn, denoted ap C, is the set of all apices of C. If a is an apex of a convex cone C, then ap C = C ∩ (2a−C). Natural generalizations of cones are given by the sets of the form K = B + C, where B is a compact convex set and C is a convex cone; these sets are called M-decomposable if C is closed, and M-predecomposable otherwise (see Goberna et al. [35] and Iusemi et al. [41], respectively). For a convex set K ⊂ Rn and a point a ∈ Rn, the generated cone Ca(K) with apex a is defined by Ca(K) = {a + λ(x−a) : x ∈ K, λ ≥ 0}. Both sets Ca(K) and cl Ca(K) are convex cones with apex a (we observe that the cone Ca(K) may be nonclosed even if K is closed). Furthermore, Ca(K) is a plane if and only if a ∈ rint K. In particular, Ca(K) = Rn if and only if a ∈ int K. The recession cone of a convex set K ⊂ Rn is defined by rec K = {e ∈ Rn : x + λe ∈ K whenever x ∈ K and λ ≥ 0}. If K is closed, then rec K is a closed convex cone with apex o, and rec K 6= {o} if and only if K is unbounded. The lineality space of K is the subspace defined by lin K = rec K ∩ (−rec K). In a standard way, the (negative) polar cone of a convex set K ⊂ Rn is defined by K◦ = {e ∈ Rn : x·e ≤ 0 for all x ∈ K}. 246 v. soltan The set K◦ is a closed convex cone with apex o. Furthermore, 1. K◦ = {o} if and only if o ∈ int K, and K◦ is a subspace if and only if o ∈ rint K. 2. K◦ = (Co(K)) ◦, (K◦)◦ = cl Co(K), lin K ◦ = K⊥. 3. If C ⊂ Rn is a closed convex cone with apex o, then a nonzero vector e ∈ Rn belongs to rint C◦ if and only if x·e < 0 for all x ∈ C \ lin C. o K◦ K Co(K) Figure 1: The polar cone of a convex set K. Given a point z in the closure of a convex set K ⊂ Rn, the polar cone (K −z)◦ is often called the normal cone of K at z and is denoted Nz(K); it consists of all vectors e ∈ Rn such that z is the nearest to e + z point in cl K. o K −z Nz(K) Figure 2: The normal cone Nz(K). The union of all normal cones Nz(K), where z ∈ cl K, is denoted nor K and called the normal cone of K. The set nor K is a cone with apex o, which is not necessarily closed or convex. One more cone associated with K is its barrier cone, defined as bar K = {e ∈ Rn : ∃γ = γ(e) ∈ R such that x·e ≤ γ for all x ∈ K}. support and separation properties 247 It is known that bar K is a convex (not necessarily closed) cone with apex o. Generally, rint (rec (cl K))◦ ⊂ nor K ⊂ bar K ⊂ (rec (cl K))◦, which implies the equalities rec (cl K) = (nor K)◦ = (bar K)◦. For instance, if K = {(x,y) : y ≥ x2}, then K◦ = {(0,y) : y ≤ 0} and rec K = {(0,y) : y ≥ 0}, (rec K)◦ = {(x,y) : y ≤ 0}, nor K = bar K = {o}∪{(x,y) : y < 0}. A point z of a convex set K ⊂ Rn is called an extreme point of K provided the set K \{z} is convex (equivalently, the equality z = (1 −λ)u + λv, where u,v ∈ K and 0 < λ < 1, is possible only if u = v = z). Similarly, z is an exposed point of K provided there is a hyperplane H ⊂ Rn satisfying the condition H ∩ K = {z}. The sets ext K and exp K of extreme and exposed points of a closed convex set K ⊂ Rn have the following properties. 4. exp K 6= ∅ ⇔ ext K 6= ∅ ⇔ K is line-free (that is, K contains no line). 5. exp K ⊂ ext K ⊂ cl (exp K). 3. Minkowski’s Theorems 3.1. Support Hyperplanes The concept of support hyperplane is at- tributed to Minkowski [63, § 8]: Given a nonempty set X ⊂ Rn, a hyperplane H ⊂ Rn supports X if it contains at least one point of X and does not cut X (that is no two points of X belong, respectively, to the opposite open halfs- paces determined by H). Clearly, the above condition “H does not cut X” can be equivalently reformulated as “X entirely lies in a closed halfspace de- termined by H.” The following well-known result is due to Minkowski (see [63, § 16] for all n ≥ 3 and [64, pp. 139 – 141] for n = 3). Theorem 3.1. ([63, § 16]) Every boundary point of a convex body K ⊂ Rn belongs to a hyperplane supporting K. Since general theory of convex sets, with credible geometric arguments, was still in rudimentary stage, Minkowski’s proof of Theorem 3.1 used an analytic description of convex bodies in terms of radial distances (see [63, § 1]). The radial distance (which later became known as the Minkowski gauge function) from a point a to a point b in Rn is a real-valued function S(a,b) satisfying the following conditions: 248 v. soltan 1. S(a,b) > 0 if a 6= b, and S(a,a) = 0; 2. if c = a + t(b−a), where t ≥ 0, then S(a,c) = tS(a,b); 3. S(a,c) ≤ S(a,b) + S(b,c) whenever a,b,c ∈ Rn. The standard surface F and the standard body K of the radial distance func- tion S(o,x) are defined, respectively, by F = {x ∈ Rn : S(o,x) = 1} and K = {x ∈ Rn : S(o,x) ≤ 1}. (3.1) Minkowski observed (see [63, § 8]) that the standard body K from (3.1) is a compact convex set containing the origin o in its interior, and, conversely, that a certain translate of a convex body can be viewed as the standard body of a suitable radial distance function. Using the properties of S(a,b), and not the convexity of K, Minkowski showed that any point of F belongs to a hyperplane supporting F (and thus supporting K). His proof consists of two steps. Step 1. Given a point z ∈ F and a scalar t > 1, there is a hyperplane H(t) contained in Rn \K and meeting the open interval (z,tz) (see Figure 3). F K o z tz H(t) Figure 3: A hyperplane H(t) in Rn \K. Step 2. Given a sequence of scalars t1, t2, . . . ( > 1) tending to 1 and a respective sequence of hyperplanes H(t1),H(t2), . . . obtained in the above Step 1 and described as β0(ti) + β1(ti)x1 + · · · + βn(ti)xn = 0, i ≥ 1, one can choose n+1 infinite subsequences βj(ti1 ),βj(ti2 ), . . . , 0 ≤ j ≤ n, which converge, respectively, to scalars β0,β1, . . . ,βn such that the limit hyperplane β0 + β1x1 + · · · + βnxn = 0 supports F at z. The analytic nature of Minkowski’s proof prompted various mathemati- cians to consider more geometric approaches. For instance, Carathéodory [16] proved Theorem 3.1 based on the following auxiliary result, afterward widely used in convex geometry (see Figure 4). support and separation properties 249 Theorem 3.2. ([16]) Let K ⊂ Rn be a convex body, and u be a point outside K. If z is the nearest to u point in K, then the hyperplane H through z orthogonal to the segment [u,z] supports K such that u and K lie in the opposite closed halfspaces determined by H. z u K H Figure 4: Illustration to Theorem 3.2. Straszewicz [86, pp. 19 – 21] gave one more method of the proof of Theorem 3.1 which uses an induction argument. Steinitz (see [83, § 11] and [84, § 26]) showed that Theorem 3.1 and Theorem 3.2 hold for the case of any proper n-dimensional convex set K ⊂ Rn, but his contribution went unno- ticed. Various proofs of Theorem 3.1 for the case of dimensions 2 and 3 were provided at that time. For instance, Brunn [10] uses a similar to Straszewicz [86] method (and later another method in [11]), while Blaschke [7, pp. 53 – 54] finds a suitable orthogonal projection of a convex body K ⊂ R3 on a plane and uses the support property of this projection in the plane. We observe that in all three sources [16, 63, 86], the resulting support hyperplane of a given convex body is obtained as the limit of a suitable con- verging sequence of hyperplanes. Proofs of Theorem 3.1 which do not employ limit procedures appeared much later. These can be found in the papers of Favard [28, Chapter 2] and Botts [9], as given below. Theorem 3.3. ([9]) If z is a boundary point of a compact convex set K ⊂ Rn, and v is a point of the unit sphere S1(z) at a largest distance from K, then the hyperplane through z orthogonal to the segment [v,z] supports K. Support properties of convex bodies are used for various classifications of their boundary points. For instance, a point z of a convex body K ⊂ Rn is called regular if it belongs to a unique support hyperplane of K. It is well- known that the set of regular points of a convex body K ⊂ Rn is everywhere 250 v. soltan dense in bd K. Historical references here are due to Jensen [44] and Bernstein [4, 5], (for the planar case), Kakeya [45], Fujiwara [31], and Reidemeister [71] (for the 3-dimensional case), and Mazur [62] (for the case of linear normed spaces of any dimension). z v K Figure 5: Illustration to Theorem 3.3. Botts [9] gave the following characteristic property of regular points: If K ⊂ Rn is a convex body, z0 ∈ bd K, and H is a hyperplane support- ing K at z0, then z0 is regular if and only if for every sequence of points z1,z2, . . . ∈ bd K \{z0} converging to z0, the sequence of numbers δ(z1,H) ‖z1 −z0‖ , δ(z2,H) ‖z2 −z0‖ , . . . tends to 0, where δ(zi,H) denotes the distance from zi to H. One more classification of boundary points derives from observing contact sets of a convex body K ⊂ Rn and its support hyperplanes. For example, the equality K = cl (conv (exp K)), proved by Straszewicz [87], implies that K has at least n + 1 exposed points, and that the set exp K is finite if and only if K is a polytope. 3.2. Bounding and Separating Hyperplanes An important class of hyperplanes with respect to a given convex body was considered by Carathéo- dory [17] and used by him to describe the closed convex hull of a compact set in Rn. Namely, a hyperplane H ⊂ Rn is said to bound a convex body K ⊂ Rn provided H ∩K = ∅. Equivalently, H bounds K if K lies in one of the open halfspaces determined by H (see Figure 6). Theorem 3.4. ([17]) If z ∈ Rn is an exterior point of a convex body K ⊂ Rn, and u is the nearest to z point in K, then the hyperplane through z orthogonal to the segment [u,z] bounds K. support and separation properties 251 The following separation theorem of Minkowski [64] originated a variety of related results. Theorem 3.5. ([64, p. 141]) If K1 and K2 are convex bodies in R3 with disjoint interiors, then there is a plane H such that the sets int K1 and int K2 belong to the opposite open halfspaces determined by H. u z K H Figure 6: Bounding hyperplane of K through a given point z. H z1 z2 K1 K2 Figure 7: Separating hyperplane of convex bodies K1 and K2. Minkowski’s proof of this theorem is divided into two cases. Case 1: K1 ∩ K2 = ∅. If points z1 ∈ K1 and z2 ∈ K2 are at a mini- mum possible distance, and if H is a plane orthogonal to the segment [z1,z2] and passing through an interior point of this segment, then K1 and K2 are contained in the opposite open halfspaces determined by H (see Figure 7). Case 2: K1 ∩K2 6= ∅. Assuming that o ∈ int K1, any smaller homothetic copy tK1 of K1, where 0 < t < 1, is disjoint from K2. By the above Case 1, there is a plane H(t) such that tK1 and K2 belong to the opposite open halfspaces determined by H(t). Given a sequence of scalars t1, t2, . . . in the interval (0, 1) which tend to 1, one can chose a sequence of planes H(ti) separating, respectively, tiK1 and K2 and converging to a plane H such that K1 and K2 belong to the opposite closed halfspaces determined by H. 252 v. soltan Interestingly, Minkowski did not use the term “separating plane”. This term (“Zwischenebene”) was used later by Brunn [12]. The assertion of Theo- rem 3.5 was formulated by Bonnesen and Fenchel [8, p. 5] for the n-dimensional case, with the term “separating hyperplane”, without any ref- erence on Minkowski [64]. Theorem 3.5 was generalized in 1936 by Eidelheit [25] for the case of convex bodies in linear normed spaces of any dimension (see the survey [51] for further bibliography). o K1 K2 Figure 8: Double cone of normal vectors to separating hyperplanes. The following result, obtained by Brunn [12], describes the normal vectors of all planes which separate a given pair of convex bodies (compare with Theorem 5.3 below). Theorem 3.6. ([12]) Let K1 and K2 be disjoint convex bodies in R3. The set of normal vectors (drawn at the origin of R3) to all planes separating K1 and K2 is a convex double cone, i.e., is the union of two opposite convex cones with the improper common apex o (see Figure 8). 3.3. Sufficient Conditions for Convexity of Solid Sets The following theorem of Minkowski [63] characterizes bounded convex surfaces in terms of their support property. Theorem 3.7. ([63, § 17]) A set F ⊂ Rn is the boundary of a convex body in Rn provided there is a point z ∈ Rn such that (a) every open halfline originated at z meets F, (b) every point of F belongs to a hyperplane supporting F . Using similar arguments, Brunn [10, p. 293] proved the theorem below. support and separation properties 253 Theorem 3.8. ([10]) If a compact set X ⊂ Rn has nonempty interior and any boundary point of X belongs to a hyperplane supporting X, then X is a convex body. Straszewicz [86, pp. 22 – 24] independently proved Theorem 3.8 under the additional assumption that X is connected and coincides with the closure of its interior. A similar result was obtained by Haalmeijer [38]: If X ⊂ Rn is the closure of an open connected set and Y is a dense subset of bd X, then X is convex provided every point x ∈ Y belongs to a hyperplane supporting X. Tietze (see [89] and [90]) gave two local versions of Theorem 3.8. Following [89], an open half-ball Qρ(z) ⊂ Rn with center z and radius ρ > 0 means the intersection of the open ball Uρ(z) and an open halfspace whose boundary hyperplane contains z. Theorem 3.9. ([89, 90]) A compact set X ⊂ Rn is convex if it satisfies any of the following two conditions: (a) X is the closure of an open connected set, and for every boundary point z of X there is an open half-ball Qρ(z), ρ = ρ(z) > 0, disjoint from X; (b) X is connected, int X 6= ∅, and there is a scalar ρ > 0 such that for every boundary point z of X a suitable open half-ball Qρ(z) is disjoint from X. X Figure 9: Illustration to Theorem 3.9. The nonconvex sets X ⊂ R2 in Figure 9 illustrate that the assumptions in both conditions (a) and (b) of Theorem 3.9 are essential. Indeed, at any boundary point z, the set X has a support open half-ball Qρ(z), where ρ = ρ(z), while int X is not connected and no constant scalar ρ > 0 satisfies condition (b). A simplified proof of part (b) of Theorem 3.9 was given by Reinhardt [70]. Also, this part was sharpened later in the following ways. 254 v. soltan 1. Gericke [34] (see also Nöbeling [67]) showed that the centers of disjoint with X half-balls Qρ(z) can be chosen in bd X\L, where L is a suitable (n− 3)-dimensional plane. 2. Süß [88] proved that condition (b) is satisfied if X is connected, has interior points, and there is a scalar ρ > 0 such that for every boundary point z of X there is a cylinder C(z) of variable height based on an (n− 1)-dimensional ball of radius ρ, with X ∩ int C(z) = ∅. 3. Schmidt (see Bieberbach [6, p. 20]) observed that it is sufficient to require the existence of a hyperplane H through z which does not meet the set X ∩Uρ(z) (without the assumption that H supports X ∩Uρ(z)). Burago and Zalgaller [13] (see pp. 395 and 415) observed that the word “compact” can be replaced by “closed” in both Theorem 3.8 and Theorem 3.9. Also, they changed the language of Theorem 3.9, replacing in (b) the requirement X ∩Qρ(z) = ∅ with the following one: X ∩Uρ(z) is supported at z by a suitable hyperplane. One more variation of Theorem 3.8 comes from geometric measure theory. Namely, a closed set X ⊂ Rn with non-empty interior is convex if and only if it has locally finite perimeter and possesses a support hyperplane at each point of its reduced boundary (see Caraballo [14, 15] for definitions and technical details). 4. Supports and Bounds of Convex Sets 4.1. Support Hyperplanes An extension of Minkowski’s definition of support hyperplane says that a hyperplane H ⊂ Rn supports a nonempty set X ⊂ Rn provided H meets its closure, cl X, and does not cut X. Analysis of the proof of Theorem 3.3 shows that its assertion holds for the case of any proper convex set K ⊂ Rn. Consequently, Theorem 3.1 can be generalized as follows: Theorem 4.1. Any boundary point of a convex set K ⊂ Rn belongs to a hyperplane supporting K. If the dimension of a convex set K ⊂ Rn is less than n, then a hyper- plane H ⊂ Rn supporting K may contain K entirely. Nevertheless, in many instances it is important to know whether H properly supports K, that is whether K 6⊂ H. Equivalent terms used in the literature are essential support (see Steinitz [84, § 28]) and nontrivial support (see Rockafellar [72, p. 100]). support and separation properties 255 Theorem 4.2. ([72, Theorem 11.6]) Let K ⊂ Rn be a convex set which is not a plane, and let F be a nonempty convex subset of cl K (for instance, F is a singleton). There is a hyperplane containing F and properly supporting K if and only if F ⊂ rbd K. It is easy to see that a hyperplane H properly supports a convex set K if and only if it meets cl K such that H ∩ rint K = ∅. The following assertion is a variation of Theorem 4.2. Theorem 4.3. ([76, Corollary 9.11]) If a plane L ⊂ Rn meets the closure of a convex set K ⊂ Rn such that L∩ rint K = ∅, then there is a hyperplane containing L and properly supporting K. The next result shows that the existence of a support hyperplane is a local property. Theorem 4.4. ([76, Problem 9.3]) Let K ⊂ Rn be a convex set, z be a point in cl K, and Bρ(z) ⊂ Rn be a closed ball of radius ρ > 0 centered at z. A hyperplane H ⊂ Rn through z supports (properly supports) K if and only if H supports (respectively, properly supports) the set K ∩Bρ(z). z K H Figure 10: Local support of K at z. If a support hyperplane H ⊂ Rn of a convex set K ⊂ Rn is described by the equation (2.1), then its normal vector e and the level scalar γ can be characterized as follows. 1. H supports (properly supports) K at a point z ∈ cl K if and only if e ∈ Nz(K) (respectively, e ∈ Nz(K) \ ort K) and γ = z·e. 2. H supports (properly supports) K if and only if e ∈ nor K (respectively, e ∈ nor K \ ort K) and γ = sup{x·e : x ∈ K}. 256 v. soltan Theorem 4.5. ([76, Theorem 12.22]) If K ⊂ Rn is a line-free convex set, then the set of nonzero vectors e ∈ Rn for which the hyperplane of the form H = {x ∈ Rn : x·e = γ}, where γ = sup{u·e : u ∈ K}, supports cl K at a single point is dense in nor K. One more assertion, independently proved by Durier [24] for the case of convex bodies and by Klee [52] for the case of line-free closed convex sets, complements Theorem 4.5 and the inclusion ext K ⊂ cl (exp K). Theorem 4.6. ([24, 52]) Let K ⊂ Rn be a line-free closed convex set, H be a hyperplane supporting K, and z be an extreme point of K that belongs to H. Then there is a sequence of points zi ∈ exp K and a respective sequence of hyperplanes Hi, i ≥ 1, satisfying the conditions: (a) Hi ∩K = {zi}, i ≥ 1, (b) z = lim i→∞ zi and H = lim i→∞ Hi. There are a few results on support properties of special convex sets. For instance: 3. If a hyperplane H ⊂ Rn supports a convex cone C ⊂ Rn, then every apex of C belongs to H. Furthermore, the intersection of all hyperplanes supporting C is precisely the apex set of cl C (see, e.g., [76], Theorem 9.43 and Theorem 9.46). 4. If a hyperplane H ⊂ Rn supports an M-predecomposable set K ⊂ Rn, expressed as the sum of a compact convex set B and a convex cone C with apex o, then H supports B (see [79]). 4.2. Bounding Hyperplanes and Halfspaces Modifying Carathéo- dory’s definition of bounding hyperplane, we say that a hyperplane H ⊂ Rn bounds a convex set K ⊂ Rn provided K is contained in a closed halfspace determined by H. Furthermore: H nontrivially bounds K if K 6⊂ H; strictly bounds K if H ∩K = ∅; and strongly bounds K if H bounds a suitable open ρ-neighborhood Uρ(K) of K. Analysis of the proof of Theorem 3.4 shows that it can be generalized in the following way. support and separation properties 257 Theorem 4.7. Let K be a proper convex set Rn and z ∈ Rn \ cl K. If u is the nearest to z point in cl K, then the hyperplane H ⊂ Rn through z orthogonal to the segment [u,z] strongly bounds K and δ(H,K) = δ(z,K) = ‖z −u‖. The results below describe all hyperplanes bounding a convex set K ⊂ Rn and having a given direction of normal vectors or containing a given point (or even a plane) of Rn. The following auxiliary statements on a hyperplane H ⊂ Rn immediately follow from definitions. 1. H bounds K if and only if H can be expressed in the form (2.1), where e ∈ bar K and sup{x·e : x ∈ K}≤ γ. 2. H properly bounds K if and only if H can be expressed in the form (2.1), where e ∈ bar K and γ satisfies both inequalities inf{x·e : x ∈ K} < γ and sup{x·e : x ∈ K}≤ γ. 3. H strictly bounds K if and only if H can be expressed in the form (2.1), where e ∈ bar K and x·e < γ for all x ∈ K. 4. H strongly bounds K if and only if H can be expressed in the form (2.1), where e ∈ bar K and sup{x·e : x ∈ K} < γ. Theorem 4.8. ([82]) For a convex set K ⊂ Rn and a point z ∈ Rn, the assertions below hold. (a) There is a hyperplane through z bounding K if and only if o /∈ int (K−z), or, equivalently, (K −z)◦ 6= {o}. (b) A hyperplane through z bounds K is and only if it can be expressed as H = {x ∈ Rn : x·e = z·e}, (4.1) where e ∈ (K −z)◦ \{o}. Theorem 4.9. ([82]) For a convex set K ⊂ Rn and a point z ∈ Rn, the assertions below hold. (a) There is a hyperplane through z properly bounding K if and only if o /∈ rint (K −z), or, equivalently, (K −z)◦ \ lin (K −z)◦ 6= ∅. (b) A hyperplane through z properly bounds K if and only if it can be expressed in the form (4.1), where e ∈ (K −z)◦ \ lin (K −z)◦. 258 v. soltan Theorem 4.10. ([82]) For a closed convex set K ⊂ Rn and a point z ∈ Rn, the assertions below hold. (a) There is a hyperplane through z strongly bounding K if and only if o /∈ cl (K −z). (b) If o /∈ cl (K − z) and e ∈ rint (K − z)◦, then the hyperplane (4.1) strongly bounds K. (c) If a hyperplane through z strongly bounds K, then it can be expressed in the form (4.1), where e ∈ (K −z)◦ \ lin (K −z)◦. If, additionally, K is compact, then e can be chosen in rint (K −z)◦. There are various extensions of Theorem 3.4 to the case of hyperplanes bounding a convex set K ⊂ Rn and containing a given plane L ⊂ Rn. 5. If L does not meet the relative interior of K, then there is a hyperplane through L properly bounding K (Rockafellar [72, Theorem 11.2]). 6. If the boundary of K does not contain a halfline and L is disjoint from cl K, then there is a hyperplane through L strictly bounding cl K (Klee [47]). 7. If δ(K,L) > 0, then there is a hyperplane H through L strongly bound- ing K such that δ(K,H) = δ(K,L) ([76, Theorem 9.6]). The families of hyperplanes through a plane L ⊂ Rn bounding (properly, strictly, or strongly) a given convex set K ⊂ Rn can be described similarly to Theorems 4.8 – 4.10. For instance, 8. There is a hyperplane which bounds K and contains L if and only if o /∈ int (K −L), or, equivalently, (K −L)◦ 6= {o}. 9. A hyperplane H ⊂ Rn bounds K and contains L if and only if H can be expressed in the form (4.1), where e ∈ (K −L)◦ \{o} and z is any point in L. We recall (see, e.g., Gale and Klee [33]) that a plane L ⊂ Rn is an asymp- tote of a set X ⊂ Rn provided cl X ∩ L = ∅ and δ(X,L) = 0. Existence of plane asymptotes is closely related to the properties of various algebraic op- erations on sets (see, e.g., Auslender and Teboulle [1] and [75], [76] for further references). For instance, the following assertions hold. 10. For a closed set X ⊂ Rn and a plane L ⊂ Rn, the sum X + L is closed if and only if there is a translate of L which is an asymptote of X. support and separation properties 259 X Figure 11: A plane asymptote of a set X. 11. For a linear transformation f : Rn → Rm and a closed set X ⊂ Rn, the set f(X) is closed if and only if there is a translate of null f which is an asymptote of X. The family of plane asymptotes of a given convex set is not hereditary. For example, if K is the convex set in R3 be given by K = { (x,y,z) : x ≥ 0, xy ≥ 1, z ≥ (x + y)2 } , then the xy- and xz-coordinate planes are the only plane asymptotes of K. Asymptotic properties of convex sets without boundary halflines are studied by Klee [49], and of cones and M-predecomposable sets in [77] and [79]. As proved in [81], every plane asymptote L of a convex set K ⊂ Rn contains a line-free closed convex cone which is an asymptote of K. The following assertions immediately follow the above definitions. 12. Any hyperplane H ⊂ Rn disjoint from a closed set X ⊂ Rn either strongly bounds X or is an asymptote of X. 13. A hyperplane H ⊂ Rn is an asymptote of a closed convex set K ⊂ Rn if and only if it can be expressed in the form (2.1), where e ∈ bar K\nor K and γ = sup{x·e : x ∈ K}. We will say that a closed halfspace V ⊂ Rn bounds a convex set K ⊂ Rn provided K ⊂ V . Furthermore, if H denotes the boundary hyperplane of V , then V supports (properly supports ) K is H supports (properly supports) K. Also, V strictly (strongly ) bounds K is H disjoint (strongly disjoint) from K. Various results on the existence or on description of different types of bounding halfspaces can be routinely derived from the above assertions on bounding hyperplanes. The results below deal with various representations of the closure of a given proper convex set K ⊂ Rn as intersections of bounding halfspaces (see [76, Chapter 9]). 260 v. soltan 14. cl K is the intersection of all closed halfspaces bounding (supporting, strictly, or strongly bounding) K. 15. If K is not a plane, X is a dense subset of rbd K, and V(X) is the family of all closed halfspaces V each properly supporting K at a point from X, then cl K = ∩(V : V ∈V(X)) and rint K = ∩(int V : V ∈V(X)). 16. If E is a dense subset of the normal cone nor K, then cl K is the inter- section of a countable family F of closed halfspaces of the form Ve(γ) = {x ∈ Rn : x·e ≤ γ}, where e ∈ E, γ ≥ sup{u·e : u ∈ K}. 17. If K is line-free, then cl K = ∩(V : V ∈G), where G denotes the family of all closed halfspaces supporting cl K at its exposed points. 5. Separation of Convex Sets 5.1. Classification of Separating Hyperplanes Various results on hyperplane separation of convex sets are usually formulated in the follow- ing terms. If K1 and K2 are convex sets in Rn and H ⊂ Rn is a hyperplane, then we say that 1. H separates K1 and K2 if K1 and K2 lie in the opposite closed halfspaces determined by H (possibly, K1 ∪K2 ⊂ H). 2. H properly separates K1 and K2 if H separates K1 and K2 such that K1 ∪K2 6⊂ H. 3. H definitely separates K1 and K2 if H separates K1 and K2 such that K1 6⊂ H and K2 6⊂ H. 4. H strictly separates K1 and K2 if H separates K1 and K2 such that both sets are disjoint from H (equivalently, K1 and K2 lie in the opposite open halfspaces determined by H). 5. H strongly separates K1 and K2 if H separates suitable open neighbor- hoods Uρ(K1) and Uρ(K2). support and separation properties 261 (b) Proper separation K1 K2 H (c) Definite separation K1 K2 H (d) Strict separation K1 K2 H (e) Strong separation K1 K2 H Figure 12: Types of hyperplane separation of convex sets. The above terminology gradually evolved in time: the term proper separa- tion is due to Rockafellar [72, p. 95], definite separation is called real by Bair and Jongmans [3], strict separation and strong separation are due to Klee [51]. The following obvious assertions provide analytical equivalences of the above definitions on separation of convex sets K1 and K2 in Rn by a hy- perplane H ⊂ Rn of the form (2.1): 1.′ H separates K1 and K2 if and only if e and γ can be chosen such that sup{x·e : x ∈ K1}≤ γ ≤ inf{x·e : x ∈ K2}. (5.1) 2.′ H properly separates K1 and K2 if and only if e and γ can be chosen to satisfy both inequalities (5.1) and inf{x·e : x ∈ K1} < sup{x·e : x ∈ K2}. (5.2) 3.′ H definitely separates K1 and K2 if and only if e and γ can be chosen to satisfy both inequalities (5.1) and inf{x·e : x ∈ K1} < γ < sup{x·e : x ∈ K2}. (5.3) 4.′ H strictly separates K1 and K2 if and only if e and γ can be chosen such that both conditions below are satisfied: u·e < inf{x·e : x ∈ K2} for every u ∈ K1, sup{x·e : x ∈ K1} < u·e for every u ∈ K2. (5.4) 262 v. soltan 5.′ H strongly separates K1 and K2 if and only if e and γ can be chosen such that sup{x·e : x ∈ K1} < γ < inf{x·e : x ∈ K2}. (5.5) The above types of separation can be refined by using asymmetric condi- tions (see Klee [51] and [78] for further details): If convex sets K1 and K2 in Rn are separated by a hyperplane H ⊂ Rn, then we say that 6. H properly separates K1 from K2 provided K1 6⊂ H. 7. H strictly separates K1 from K2 provided K1 is disjoint from H. The relation between symmetric and asymmetric types of separation is described in the following theorem. Theorem 5.1. ([76, Theorem 10.6]) If K1 and K2 are convex sets and H1 and H2 are hyperplanes such that Hi properly (strictly) separates Ki from K3−i, i = 1, 2, then there is a hyperplane containing H1 ∩ H2 and properly (strictly) separating K1 and K2. The results below describe the hyperplanes which separate a pair of convex sets and have a given direction of normals or contain a given point. An initial step towards description of separating hyperplanes with a given direction of normals consists in reduction to the case of a single convex set. Theorem 5.2. ([76, Theorem 10.7]) Let K1 and K2 be convex sets in Rn, and H be a hyperplane of the form (2.1). Then the assertions below hold. (a) A translate of H separates (properly separates) K1 and K2 if and only if the subspace (2.2) bounds (properly bounds) K1 −K2. (b) A translate of H strictly separates at least one of the sets K1 and K2 from the other if and only if the subspace (2.2) strictly bounds K1−K2. (c) A translate of H strongly separates K1 and K2 if and only if the subspace (2.2) strongly bounds K1 −K2. A combination of Theorems 4.8 – 4.10 and Theorem 5.2 implies the following assertions. Theorem 5.3. ([78, 82]) Given convex sets K1 and K2 in Rn, the fol- lowing assertions hold. support and separation properties 263 (a) There is a hyperplane separating K1 and K2 if and only if any of the following equivalent conditions holds: int K1 ∩ int K2 = ∅, o /∈ int (K1 −K2), (K1 −K2)◦ 6= {o}. Furthermore, a translate of a hyperplane H ⊂ Rn separates K1 and K2 if and only if H can be expressed in the form (2.1), where e ∈ (K1 −K2)◦ \{o}. (b) There is a hyperplane properly separating K1 and K2 if and only if any of the following equivalent conditions holds: rint K1 ∩ rint K2 = ∅, o /∈ rint (K1 −K2), (K1 −K2)◦ \ lin (K1 −K2)◦ 6= ∅. Furthermore, a translate of a hyperplane H ⊂ Rn properly separates K1 and K2 if and only if H can be expressed in the form (2.1), where e ∈ (K1 −K2)◦ \ lin (K1 −K2)◦. (c) There is a hyperplane definitely separating K1 and K2 if and only if either o /∈ cl (K1 −K2) or o ∈ cl (K1 −K2) and (K1 −K2)◦ \ (ort K1 ∪ ort K2) 6= ∅. Furthermore, a translate of a hyperplane H ⊂ Rn definitely separates K1 and K2 if and only if H can be expressed in the form (2.1) such that one of the following conditions is satisfied: (i) o /∈ cl (K1 −K2) and e ∈ rint (K1 −K2)◦ ∪ (rbd (K1 −K2)◦ \ (ort K1 ∪ ort K2)), (ii) o ∈ cl (K1 −K2) and e ∈ (K1 −K2)◦ \ (ort K1 ∪ ort K2). (d) There exists some hyperplane strongly separating K1 and K2 if and only if o /∈ cl (K1 − K2). Furthermore, if o /∈ cl (K1 − K2) and e ∈ rint (K1 − K2)◦, then a suitable translate of a hyperplane of the form (2.1) strongly separates K1 and K2. A variation of Theorem 5.3 allows us to describe all hyperplanes which separate a pair of convex sets and contain a given point of Rn. This description uses the following auxiliary lemma. 264 v. soltan Lemma 5.4. ([82]) Convex sets K1 and K2 in Rn are separated (properly separated) by a hyperplane H ⊂ Rn through a given point z ∈ Rn if and only if the generated cones Cz(K1) and Cz(K2) are separated (properly separated) by H. Theorem 5.5. ([82]) Given convex sets K1 and K2 in Rn and a point z ∈ Rn, let D1 = Cz(K1) −z, D2 = Cz(K2) −z. The assertions below hold. (a) There is a hyperplane through z separating K1 and K2 if and only if the cones D1 and D2 satisfy any of the following equivalent conditions: o /∈ int (D1 −D2), (D1 −D2)◦ 6= {o}, D◦1 ∩ (−D ◦ 2) 6= {o}. (b) There is a hyperplane through z properly separating K1 and K2 if and only if the cones D1 and D2 satisfy any of the following equivalent conditions: o /∈ rint (D1 −D2), (D1 −D2)◦ is not a subspace, D◦1 ∩ (−D ◦ 2) is not a subspace. 5.2. Geometric Conditions on Hyperplane Separation Theorem 5.3 provides a unified description of all hyperplanes separating convex sets K1 and K2 in Rn, which is formulated in terms of the polar cone (K1 − K2)◦. Some other types of geometric conditions that guaranty the existence of a desired type of separation are given below. Proper separation. The condition rint K1 ∩ rint K2 = ∅ was already men- tioned in Theorem 5.3. It was obtained in various forms of generality by Fenchel [30, p. 48], Klee [46], and Rockafellar [72, Theorem 11.3]. A related result on asymmetric type of proper separation is due to Rock- afellar [72, Theorem 20.2]: Given convex sets K1 and K2 in Rn such that K2 is polyhedral, there is a hyperplane properly separating K1 from K2 if and only if rint K1 ∩K2 = ∅. We observe that the latter assertion does not hold if the set K2 is not polyhedral. For instance, if K1 and K2 are planar circular disks in R3 given by K1 = {(x,y, 0) : x2 + (y − 1)2 ≤ 1}, K2 = {(0,y,z) : y2 + (z − 1)2 ≤ 1}, then rint K1 ∩K2 = ∅, while K1 is not properly separated from K2. support and separation properties 265 Strict separation. The only known geometric result on strict separation of convex sets is attributed to Klee [47]: If none of the disjoint closed convex sets K1 and K2 in Rn is a plane, and none of the sets rbd K1 and rbd K2 contains a halfline, then K1 and K2 are strictly separated by a hyperplane. Strong separation. An obvious continuity argument shows that if convex sets K1 and K2 are strongly separated by a hyperplane H, then there is a slab of positive width which separates K1 and K2 and whose boundary hyperplanes are parallel to H. A natural question here is to determine the maximum possible width of such a slab. The answer to this question was given by Dax [20] (see also [76, Theorem 10.20]). H K1 K2 Figure 13: Strict separation of K1 and K2. δ(K1,K2) K2 K1 H1 H2 Figure 14: Separation of K1 and K2 by a slab of maximum width. Theorem 5.6. ([20]) If convex sets K1 and K2 in Rn are strongly disjoint (that is, δ(K1,K2) > 0), then there is a unique pair of parallel hyperplanes H1 and H2 in Rn, both separating K1 and K2 and satisfying the condition δ(H1,H2) = δ(K1,K2). The above equality δ(H1,H2) = δ(K1,K2), without specifying the unique- ness of the pair {H1,H2}, was obtained later by Gabidullina [32] for the case when at least one of the sets K1 and K2 is compact. A similar question on strong separation of convex sets K1 and K2 concerns the existence of a pair of nearest points z1 ∈ cl K1 and z2 ∈ cl K2. A simple 266 v. soltan geometric argument shows that in this case the hyperplanes through z1 and z2 orthogonal to [z1,z2] form a slab of maximum width separating K1 and K2. A sufficient condition for the existence of such a pair {z1,z2} can be found in [76]. z1 z2 K1 K2 Figure 15: A nearest pair of points in K1 and K2. Theorem 5.7. ([76, Theorem 10.24]) If convex sets K1 and K2 in Rn sat- isfy the condition rec (cl K1) ∩ rec (cl K2) = {o}, then δ(K1,K2) = ‖z1 − z2‖ for suitable points z1 ∈ cl K1 and z2 ∈ cl K2. In particular, a nearest pair {z1,z2} exists provided at least one of the sets K1 and K2 is bounded. A useful result on strong separation of convex sets was obtained by De Wilde [21]. Theorem 5.8. ([21]) If K1 and K2 are disjoint closed convex sets in Rn, then the following conditions are equivalent: (a) Both K1 and K2 are line-free and rec K1 ∩ rec K2 = {o}. (b) There are parallel disjoint hyperplanes H1 and H2 both separating K1 and K2 such that Hi ∩Ki is an exposed point of Ki, i = 1, 2. Analysis of the proof of Theorem 5.8 shows that the exposed points H1∩K1 and H2∩K2 are not necessarily the nearest. Nevertheless, H1 and H2 may be chosen to satisfy the condition δ(H1,H2) > δ(K1,K2)−ε for any given scalar ε > 0. Maximal separation. Klee [50] obtained various results regarding strict and strong separation of convex sets by hyperplanes. Given a pair {F,G} of non- empty families of closed convex sets in Rn, we say that F is maximal with respect to a certain type S of separation provided it satisfies the following conditions: 1. The sets F and G are S-separated whenever F ∈ F and G ∈ G, with F ∩G = ∅. support and separation properties 267 2. For every F ∈F, there is G ∈G such that F ∩G = ∅. 3. For every closed convex set F /∈F, there is G ∈G such that F ∩G = ∅ but F and G are not S-separated. Following Gale and Klee [33], we say that a closed convex set K ⊂ Rn is continuous provided K admits no boundary halfline and no line asymptote. Given disjoint closed convex sets F and G in Rn, the assertions below hold (see [50]). 4. Each of the following conditions implies that F and G are strictly sep- arated and represents a maximal theorem for strict separation: (a) F is continuous, (b) neither F nor G admits a line asymptote, (c) neither F nor G has a boundary halfline. 5. Each of the following conditions implies that F and G are strongly sep- arated and represents a maximal theorem for strong separation: (d) F is continuous, (e) neither F nor G admits a line asymptote. An extensive development and generalization of Klee’s results on maximal separation is given in the book of Fajardo, Goberna, Rodŕıguez, and Vicente- Pérez [27]. Theorem 1.2 and Theorem 1.3 from this book give a comprehensive list of various maximal separation assertions for the case of evenly convex sets. (According to Fenchel [29], a convex set in Rn is called evenly convex if it is the intersection of a family of open halfspaces. It is easy to see that every proper closed convex sets is evenly convex.) 5.3. Sharp Separation of Convex Cones If convex cones C1 and C2 with a common apex in Rn are separated by a hyperplane H ⊂ Rn, then H supports both cones cl C1 and cl C2. Consequently, ap (cl C1)∪ap (cl C2) ⊂ H. In this regard, we will say that H sharply separates C1 from C2 provided H ∩ cl C1 = ap (cl C1). Similarly, H sharply separates C1 and C2 if H ∩ (cl C1) = ap (cl C1) and H ∩ cl C2 = ap (cl C2). The next two theorems give criteria for sharp separation of cones in terms of their polar cones. 268 v. soltan Theorem 5.9. ([78]) If C1 and C2 are convex cones in Rn with a common apex a ∈ Rn, then the following conditions are equivalent. (a) C1 is sharply separated from C2. (b) The set E = rint (C1 −a)◦ ∩ (a−C2)◦ has positive dimension. Theorem 5.10. ([74, 78]) If C1 and C2 are convex cones in Rn with a common apex a ∈ Rn, then the following conditions are equivalent. (a) C1 and C2 are sharply separated. (b) Each of the cones C1 and C2 is sharply separated from the other. (c) The set D = rint (C1 −a)◦ ∩ rint (a−C2)◦ has positive dimension. Analysis of the proof of Theorem 5.10 reveals a simple corollary: If C1 is not a plane and is sharply separated from C2, then C1 is properly separated from C2. The converse assertion is not true. For instance, in R2, the cone C1 = {(x, 0) : x ∈ R} is separated sharply but not properly from the cone C2 = {(x,y) : 0 ≤ x, 0 ≤ y ≤ x}, while C2 is separated properly but not sharply from C1 (see Figure 16). C2C1 Figure 16: Proper but not sharp separation of cones C1 and C2. A geometric criterion for sharp separation of convex cones is given in the following theorem. Theorem 5.11. ([74]) Let C1 and C2 be convex cones in Rn with a com- mon apex a ∈ Rn. The conditions below are equivalent. (a) C1 and C2 are sharply separated by a hyperplane. (b) cl C1 ∩ cl C2 = ap (cl C1) ∩ ap (cl C2) and at least one of the cases below holds: (i) dim (C1 ∪C2) 6 n− 1, (ii) at least one of the cones C1 and C2 is not a plane. support and separation properties 269 In terms of continuous linear functionals on a linear topological space, The- orem 5.11, formulated for the case of closed convex cones with a common apex o, was proved earlier by Klee [46] under the assumption ap C1 ∩ ap C2 = {o}, and by Bair and Gwinner [2] under the condition that ap C1 ∩ ap C2 is a subspace. 5.4. Penumbras and Separation Following Rockafellar [72, p. 22], we recall that the penumbra of a convex set K1 with respect to another convex set K2, denoted below P(K1,K2), is defined by P(K1,K2) = ∪(µK1 + (1 −µ)K2 : µ ≥ 1) = {µx1 + (1 −µ)x2 : µ ≥ 1,x1 ∈ K1,x2 ∈ K2}. Geometrically, P(K1,K2) is the union of all closed halflines initiated at the points of K1 in the directions of vectors from K1 − K2 (see Fig. 17). It is possible to show (see [80]) that both sets P(K1,K2) and P(K2,K1) are convex and contain K1 and K2, respectively. The following theorem illustrates the role of penumbras in separation of convex sets. P(K2,K1) K2 K1 P(K1,K2) H Figure 17: Illustration to Theorem 5.12. Theorem 5.12. ([80]) Let K1 and K2 be convex sets in Rn. A hyper- plane H ⊂ Rn separates (respectively, properly, strictly, or strongly) K1 and K2 if and only if it separates (respectively, nontrivially, strictly, or strongly) the sets P(K1,K2) and P(K2,K1). Given convex sets K1 and K2 in Rn, denote by H1(K1,K2) (respectively, by H2(K1,K2) and H3(K1,K2)) the family of all hyperplanes properly (re- spectively, strictly and strongly) separating K1 and K2. Also, let Ei(K1,K2) = ∪(H : H ∈Hi), i = 1, 2, 3. 270 v. soltan Theorem 5.13. ([80]) If convex sets K1 and K2 in Rn satisfy the condi- tion rint K1 ∩ rint K2 = ∅, then E1(K1,K2) = Rn \ (rint P(K1,K2) ∪ rint P(K2,K1)). Furthermore, a hyperplane H ⊂ Rn properly separates K1 and K2 if and only if H ⊂ E1(K1,K2) and H ∩ aff (K1 ∪K2) 6= ∅. Corollary 5.14. ([80]) If convex sets K1 and K2 in Rn satisfy the con- dition cl K1 ∩ cl K2 = ∅, then E2(K1,K2) ⊂ F2(K1,K2), where F2(K1,K2) = Rn \ (P(cl K1, cl K2) ∪P(cl K2, cl K1))). Furthermore, a hyperplane H ⊂ Rn strictly separates cl K1 and cl K2 if and only if H ⊂ F2(K1,K2) and H ∩ aff (K1 ∪K2) 6= ∅. The inclusion E2(K1,K2) ⊂ F2(K1,K2) in Corollary 5.14 may be proper. Indeed, consider the closed convex sets K1 = {(x, 1) : 0 ≤ x ≤ 1} and K2 = {(x, 0) : x ∈ R}. Then E2(K1,K2) = {(x,y) : 0 < y < 1}, while F2(K1,K2) = E2(K1,K2) ∪{(x, 1) : x < 0}∪{(x, 1) : x > 1}. Theorem 5.15. ([80]) If convex sets K1 and K2 in Rn are strongly dis- joint, then E3(K1,K2) = Rn \ (cl P(K1,K2) ∪ cl P(K2,K1)). The following assertions from [80] relate various properties of penumbras to some known classes of convex sets in Rn. 1. If K1 is compact, then P(K1,K2) is an M-predecomposable set. 2. If both K1 and K2 are compact and K1 ∩K2 = ∅, then P(K1,K2) is an M-decomposable set. 3. If both K1 and K2 are polyhedra, then cl P(K1,K2) is a polyhedron. 4. If both K1 and K2 are polytopes, then P(K1,K2) is a polyhedron. support and separation properties 271 5.5. Hemispaces The following concept was introduced by Motzkin [66, Lecture III] in three dimensions, and, independently, by Hammer [39] in vector spaces of any dimension: Given a point v ∈ Rn, any maximal (under inclusion) convex subset of Rn\{v}, denoted Sv, is called a semispace of Rn at v (in [43] and [53] these sets are called hypercones ). The next properties of semispaces can be easily obtained (see [39, 43, 66]). 1. For a semispace Sv ⊂ Rn, both sets Sv and Rn \ Sv are convex cones with apex v. 2. For a convex set K⊂Rn\{v}, there is a semispace Sv⊂Rn containing K. 3. If C ⊂ Rn is a convex cone with improper apex v ∈ Rn and B ⊂ Rn is a convex set missing v and disjoint from C, then there is a semispace Sv ⊂ Rn containing C and disjoint from B (Jamison [43] for the case v = o). Additional properties of semispaces in vector spaces of any dimension can be found in the papers [22, 43, 48, 56, 65]. Theorem 5.16. ([39]) The family of all semispaces of Rn is the smallest among all families F of convex sets in Rn satisfying the following condition: every proper convex set K ⊂ Rn is the intersection of some elements from F. The structure of semispaces can be described in different ways. The first one, briefly mentioned by Hammer [40] (see the books [58, Satz 1.10], and [76, Theorem 10.32] for complete proofs), uses a nested family of planes {v} = L0 ⊂ L1 ⊂ ···⊂ Ln−1 ⊂ Ln = Rn, dim Li = i, 0 ≤ i ≤ n, (5.6) and their halfplanes E1, . . . ,En, where Ei an open halfplane of Li determined by Li−1, 1 ≤ i ≤ n. Theorem 5.17. ([40]) If Sv ⊂ Rn is a semispace at v ∈ Rn, then there is a nested sequence of planes of the form (5.6) and a respective sequence of open halfplanes E1, . . . ,En such that Sv = E1 ∪·· ·∪En. Conversely, any set of the form E1 ∪·· ·∪En is a semispace at v. Another way (given by Hammer [39, 40] without proof) is based on the choice of a suitable basis for Rn. Theorem 5.18. ([39, 40]) If Sv ⊂ Rn is a semispace at v ∈ Rn, then there is a basis e1, . . . ,en for Rn such that Sv consists of all vectors of the form 272 v. soltan v + α1e1 + · · ·+ αnen, where α21 + · · ·+ α 2 n > 0 and the first nonzero scalar in the sequence α1, . . . ,αn is positive. Conversely, given any basis e1, . . . ,en for Rn, the set of described above vectors is a semispace at v. The equivalence of description of semispaces in Theorem 5.17 and Theorem 5.18 follows from the simple geometric arguments: 4. If e1, . . . ,en is a basis for Rn, then the open halfplanes Ei from Theorem 5.17 can be chosen as Ei = v +{αn−i+1en−i+1 +· · ·+αnen : αn−i+1 > 0}, 1 ≤ i ≤ n. (5.7) 5. For any choice of planes (5.6) and of respective halfplanes E1, . . . ,En, nonzero vectors ei ∈ (En−i+1 −v) \ (Ln−i −v), 1 ≤ i ≤ n, form a basis for Rn such that the equalities (5.7) hold. Independently, Mart́ınez-Legaz [59] described a similar separation result, based on lexicographic ordering � of Rn. We recall that for distinct vectors x = (x1, . . . ,xn) and y = (y1, . . . ,yn), one can write x ≺ y if xi < yi, with i being the first index in {1, . . . ,n} for which xi 6= yi; also, x � y if x ≺ y or x = y. In a standard way, a invertible n × n matrix A is orthogonal if A−1 = AT . Theorem 5.19. ([59]) For a proper convex set K ⊂ Rn and a point x0 /∈ K, there is an invertible (even orthogonal) n×n matrix A and a vector v ∈ Rn such that Ax ≺ v � Ax0 whenever x ∈ K. Although Mart́ınez-Legaz [59] made an observation that the sets from Theorem 5.19 are similar in their properties to semispaces, it was Singer [73] who proved the following assertion: A set M ⊂ Rn is a semispace at v if and only if there is an invertible matrix n×n matrix A such that M = {x ∈ Rn : Ax ≺ v}. The result below is proved by Tukey [91] (the condition that the vector space E should be normed is superfluous) and, independently, by Stone [85] (see Theorem 7 from Chapter 3). Theorem 5.20. ([85, 91]) Any pair of disjoint convex sets K1 and K2 in a vector space E can be separated by complementary convex sets Q1 and Q2: K1 ⊂ Q1, K2 ⊂ Q2, Q1 ∪Q2 = E, Q1 ∩Q2 = ∅. support and separation properties 273 We observe that Theorem 5.20 cannot be extended to the case of more than two convex sets. For instance, the convex cones C1,C2, and C3 in the plane, depicted in Figure 18, cannot be enlarged into pairwise disjoint (even pairwise non-overlapping) convex sets whose union is the entire plane. C1 C2 C3 Figure 18: No convex extensions of cones C1,C2 and C3 cover the whole plane. The following results are similar to those from Theorem 5.20. 6. If C ⊂ Rn is a convex cone with apex o such that C∩(−C) = {o}, then there is a convex cone C′ with apex o satisfying the conditions C ⊂ C′, C′ ∩ (−C′) = {o}, and C′ ∪ (−C′) = Rn (Ellis [26]). 7. Let F ba a commuting family of affine transformations in Rn, and let K1 and K2 be disjoint convex sets both invariant with respect to trans- formations from F. Then there are complementary F-invariant convex sets Q1 and Q2 such that K1 ⊂ Q1 and K2 ⊂ Q2 (Páles [68]). Following Jamison [42], we say that a proper convex subset Q of Rn is a hemispace provided its complement Rn \Q is a convex set. In [54] and [76], hemispaces are also called convex halfspaces. Various properties of hemispaces in vectors spaces of any dimension can be found in the papers [23, 42, 55] (see [36] for related material). A description of hemispaces in Rn, similar to that of Theorem 5.18, was obtained by Lassak [54]. Theorem 5.21. ([54]) If Q and Q′ are complementary hemispaces in Rn, then there is a point v ∈ Rn, a (orthogonal) basis e1, . . . ,en for Rn, and an integer r ≥ 1 such that one of the sets Q and Q′ consists of all vectors of the form v +αrer +· · ·+αnen, where α2r +· · ·+α2n > 0 and the first nonzero scalar in the sequence αr, . . . ,αn is positive. Conversely, for any choice of a point v ∈ Rn, a basis e1, . . . ,en for Rn, and an integer r ≥ 1, the sets of vectors described above is a hemispace in Rn. 274 v. soltan The next theorem shows that the above description of complementary hemispaces can be reformulated in terms of nested sequences of planes (5.6) and of their halfplanes E1, . . . ,En. Theorem 5.22. ([76, Theorem 10.28]) If Q and Q′ are complementary hemispaces in Rn, then there is a sequence of planes of the form (5.6) and an integer 1 6 r 6 n such that either Q = Fr and Q ′ = F ′r, or Q = F ′ r and Q′ = Fr, with Fr = Er ∪·· ·∪En, F ′r = Lr−1 ∪E ′ r ∪·· ·∪E ′ n, 1 6 r 6 n, (5.8) where Ei,E ′ i are complementary open halfplanes of Li determined by Li−1. Independently, Mart́ınez-Legaz [59] defined a hemispace in Rn as the set of vectors x ∈ Rn satisfying the condition Ax ≺ v, where A is an arbitrary (not necessarily invertible) n×n matrix and v ∈ Rn. 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Tukey, Some notes on the separation of convex sets, Portugal. Math. 3 (1942), 95 – 102. Introduction Preliminaries Minkowski's theorems Support hyperplanes Bounding and separating hyperplanes Sufficient conditions for convexity of solid sets Supports and bounds of convex sets Support hyperplanes Bounding hyperplanes and halfspaces Separation of convex sets Classification of separating hyperplanes Geometric conditions on hyperplane separation Sharp separation of convex cones Penumbras and separation Hemispaces