� EXTRACTA MATHEMATICAE Volumen 33, Número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura EXTRACTA MATHEMATICAE Vol. 36, Num. 1 (2021), 81 – 98 doi:10.17398/2605-5686.36.1.81 Available online December 10, 2020 Cone asymptotes of convex sets V. Soltan Department of Mathematical Sciences, George Mason University 4400 University Drive, Fairfax, VA 22030, USA vsoltan@gmu.edu Received November 18, 2020 Presented by Horst Martini Accepted November 28, 2020 Abstract: Based on the notion of plane asymptote, we introduce the new concept of cone asymptote of a set in the n-dimensional Euclidean space. We discuss the existence and describe some families of cone asymptotes. Key words: Plane asymptote, cone asymptote, convex set. MSC (2020): 52A20, 90C25. 1. Introduction Originated in geometry, a widely used definition of an asymptote says that it is a line (or a halfline) that continually approaches a curve or a surface but does not meet it. In convex geometry, this definition was generalized by Gale and Klee [2], who defined an asymptote of a nonempty closed subset X of the Euclidean space Rn as a halfline h which lies in Rn \X and satisfies the condition δ(X,h) = 0, where the inf -distance δ(X,Y ) between nonempty sets X and Y in Rn is given by δ(X,Y ) = inf{‖x−y‖ : x ∈ X, y ∈ Y}. Later, Klee [4] introduced the concept of j-asymptote of a closed convex set K ⊂ Rn, as a plane L ⊂ Rn of dimension j, 1 ≤ j ≤ n − 1, satisfying the conditions K ∩L = ∅ and δ(K,L) = 0. In particular, a 1-asymptote is a line asymptote (see Figure 1). Plane asymptotes appeared to be a useful tool in the study of various properties of convex sets. For instance, closed convex sets in Rn without plane asymptotes are precisely those whose affine images are closed (see [1, 5, 9]). Certain classes of convex sets (like continuous convex sets, M-decomposable sets, and M-polyhedral sets) have suitable geometric properties due to the ISSN: 0213-8743 (print), 2605-5686 (online) c©The author(s) - Released under a Creative Commons Attribution License (CC BY-NC 3.0) https://doi.org/10.17398/2605-5686.36.1.81 mailto:vsoltan@gmu.edu https://publicaciones.unex.es/index.php/EM https://creativecommons.org/licenses/by-nc/3.0/ 82 v. soltan K L Figure 1: A plane asymptote L of a convex set K. absence of plane asymptotes (see [2, 4, 7, 8, 11]). The range of dimensions of plane asymptotes of a given closed convex set in Rn was investigated in [3, 5]. In the study of asymptotic behavior of convex sets, it is often desirable to describe their line or halfline asymptotes. This task is often difficult (if possible at all) partly due to the fact that the family of plane asymptotes is not hereditary (see, for instance, Example 3.5 below). Our goal here is to show that plane asymptotes of a given convex set K ⊂ Rn always contain closed cones which are asymptotes of K (see Theorem 3.3 and Theorem 4.1 below). Also, we study some geometric properties of cone asymptotes. For terminological convenience, we will say that a nonempty closed set X ⊂ Rn is an asymptote of a nonempty closed set Y ⊂ Rn provided X∩Y = ∅ and δ(X,Y ) = 0. Furthermore, we use the expression cone asymptote to distinguish it from another established concept, asymptotic cone CX of a nonempty set X ⊂ Rn, which is a cone with apex at the origin o, defined by CX = ⋂ ( cl((0,ε]X) : ε > 0 ) . We conclude this section with necessary definitions, notation, and results on convex sets in the n-dimensional Euclidean space Rn (see, e.g., [10] for details). The elements of Rn are called vectors, or points. In what follows, o stands for the zero vector of Rn. We denote by [u,v], and [u,v〉, respectively, the closed segment with endpoints u,v ∈ Rn and the closed halfline through v with endpoint u: [u,v] = {(1 −λ)u + λv : 0 ≤ λ ≤ 1}, [u,v〉 = {(1 −λ)u + λv : λ ≥ 0}. Also, u ·v will mean the dot product of u and v, and ‖u‖ denotes the norm of u. Given a pair of subspaces T ⊂ S of Rn, a subspace F ⊂ Rn will be called complementary to T in S provided F ∩ T = {o} and F + T = S. By an r-dimensional plane L in Rn, where 0 ≤ r ≤ n, we mean a translate of cone asymptotes of convex sets 83 a suitable r-dimensional subspace SL of Rn: L = u + SL, where u ∈ Rn. The subspace SL is uniquely determined by L and equals L−v for any choice of v ∈ L. A closed halfspace V of Rn is defined by V = {x ∈ Rn : x·e ≤ γ}, where e is a nonzero vector and γ is a scalar. If L ⊂ Rn is a plane of positive dimension, then by a closed halfplane of L we will mean any set of the form L∩V , where V is a closed halfspace of Rn satisfying the condition ∅ 6= L∩V 6= L. In what follows, K means a nonempty convex set in Rn. The relative in- terior and relative boundary of K are denoted rint K and rbd K, respectively. A convex set K ⊂ Rn is called line-free if it contains no lines. For a nonempty set X ⊂ Rn, the notations cl X and conv X stand, respectively, for the closure and the convex hull of X. If X is bounded, then the set Y = conv(cl X) is compact. We recall that a nonempty set C in Rn is a cone with apex v ∈ Rn if v + λ(x − v) ∈ C whenever λ ≥ 0 and x ∈ C. (Obviously, this definition implies that v ∈ C, although a stronger condition λ > 0 can be beneficial; see, e.g., [6].) A cone C with apex v is called convex if it is a convex set, and is called nontrivial if C 6= {v}. The polar cone C◦ of a convex cone C ⊂ Rn with apex o is defined by C◦ = {x ∈ Rn : x·e ≤ 0 ∀e ∈ C}. It is known that C◦ is a closed convex cone with apex o. Furthermore, C◦ is n-dimensional provided C is line-free. For a convex set K ⊂ Rn and a point v ∈ Rn, the generated cone Cv(K) is defined by Cv(K) = {v + λ(x−v) : x ∈ K, λ ≥ 0}. Both sets Cv(K) and cl Cv(K) are convex cones with apex v. 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}, and the lineality space of K is the subspace defined by lin K = rec K ∩ (−rec K). We will use the following properties of recession cones and lineality spaces of closed convex sets. (P1) rec K is a closed convex cone with apex o; it is distinct from {o} if and only if K is unbounded. 84 v. soltan (P2) For any given point u ∈ K, the cone rec K is the largest among all closed convex cones C ⊂ Rn with apex o satisfying the inclusion u + C ⊂ K. (P3) One has K + lin K = K. (P4) If C ⊂ Rn is a closed convex cone with apex o, then lin C is the largest subspace contained in C. 2. Halfplane asymptotes Klee [4] (also see [3]) observed that a line asymptote of a closed convex set K ⊂ Rn contains a halfline asymptote. Our first result partly generalizes this assertion to the case of halfplanes of a plane asymptote. Theorem 2.1. If K ⊂ Rn is a closed convex set and L ⊂ Rn is a plane asymptote of K, then the following assertions hold. (a) For any bounded set X ⊂ L, there is a closed halfplane of L, disjoint from X, which is an asymptote of K. (b) For any bounded set Z ⊂ L, there is a closed halfplane of L, containing Z, which is an asymptote of K. Proof. (a) Consider the sets Mr = {x ∈ L : δ(K,x) ≤ 1/r}, r ≥ 1. We state that every set Mr is nonempty, closed, and convex. Indeed, since δ(K,L) = 0, there are points p ∈ K and q ∈ L such that ‖p − q‖ ≤ 1/r. Obviously, δ(K,q) ≤ ‖p − q‖ ≤ 1/r, which gives the inclusion q ∈ Mr. If a sequence of points q1,q2, . . . ∈ Mr converges to a point q, then q ∈ L due to the closedness of L. Because the function δ(K,x) continuously depends on x ∈ Rn (see, e.g., [10, Theorem 8.21]), one has δ(K,q) = lim i→∞ δ(K,qi) ≤ 1/r, implying the inclusion q ∈ Mr. So, the set Mr is closed. For the convexity of Mr, choose any points q,q ′ ∈ Mr and a scalar ε > 0. There are points p,p′ ∈ K such that ‖p−q‖≤ δ(K,q) + ε ≤ 1/r + ε, ‖p′ −q′‖≤ δ(K,q′) + ε ≤ 1/r + ε. cone asymptotes of convex sets 85 Given a scalar λ ∈ [0, 1], put p0 = (1 −λ)p + λp′ and q0 = (1 −λ)q + λq′. Then p0 ∈ K by the convexity of K, and q0 ∈ L because (1 −µ)q + µq′ ∈ L whenever µ ∈ R (see [10, Theorem 2.38]). Furthermore, δ(K,q0) ≤‖p0 −q0‖ ≤ (1 −λ)‖p−q‖ + λ‖p′ −q′‖≤ 1/r + ε for each ε > 0. Thus δ(K,q0) ≤ 1/r, which gives q0 ∈ Mr. So, the set Mr is convex. Next, we observe that M1 ∩M2 ∩·· · = ∅. Indeed, assuming the existence of a point u ∈ M1 ∩M2 ∩·· · ⊂ L, we would have δ(K,u) ≤ inf{1/r : r ≥ 1} = 0, which gives u ∈ K. The latter contradicts the assumption K ∩L = ∅. Let X be a bounded subset of L. Then the convex set Y = conv(cl X) is compact. By the above argument, Y ∩M1∩M2∩·· · = ∅. Since the sequence of closed sets M1,M2, . . . is decreasing, the compactness of Y implies the existence of an integer t such that Y ∩ Mr = ∅ for all r ≥ t. Furthermore, δ(Y,Mt) > 0. Hence there is a hyperplane H ⊂ Rn separating Y and Mt such that these sets are contained in the opposite open halfspaces determined by H (see, e.g., [10, Theorem 10.12]). Denote by V the closed halfspace determined by H and containing Mt, and let P = L∩V . Clearly, P is a closed halfplane of L containing Mt and disjoint from Y . Furthermore, δ(K,P) ≤ inf{δ(K,Mr) : r ≥ t}≤ inf{1/r : r ≥ t} = 0. Summing up, P is an asymptote of K. (b) Let Z be a bounded subset of L, and let P be a closed halfplane of L which is an asymptote of K (the existence of P is proved above). By a compactness argument, one can choose a translate v + P of P which lies in L and contains P ∪Z. Obviously, the halfplane v + P is an asymptote of K. We observe that the method of proof of Theorem 2.1 gives a limited choice of halfplane asymptotes of K, as illustrated by the next example. Example 2.2. Consider K = {(x,y,z) : x > 0,y ≥ 1/x}, closed convex set in R3. It is easy to see that the coordinate xz-plane of R3, say L, is an asymptote of K. The sets Mr ⊂ L defined in Theorem 2.1 are vertical closed 86 v. soltan halfplanes of L of the form {(x, 0,z) : x ≥ αr}, r ≥ 1. Consequently, the halfplane asymptote P of K described in Theorem 2.1 should be of a similar form. On the other hand, it is easy to see that the only closed halfplanes of L, which are not asymptotes of K, have the form {(x, 0,z) : x ≤ λ}. 3. Cone asymptotes We are going to refine the argument of Theorem 2.1 to describe a wider family of asymptotes which are subsets of a given plane asymptote. For a nontrivial closed convex cone C ⊂ Rn with apex o, consider its conic ε- neighborhood Dε(C), ε > 0, defined as the union of all closed halflines h ⊂ Rn with apex o which form with C an angle of size at most ε. Clearly, Dε(C) is a closed cone (not necessarily convex) with apex o. For the following lemma, we recall that SL denotes the subspace which is a translate of a plane L ⊂ Rn. Lemma 3.1. If L ⊂ Rn is a plane asymptote of a closed convex set K ⊂ Rn, then {o} 6= rec K ∩SL 6⊂ lin K. Furthermore, rec K ∩SL 6= SL. Proof. The assertion {o} 6= rec K ∩ SL 6⊂ lin K is proved in [9]. Next, assume for a moment that rec K ∩ SL = SL. Then SL ⊂ rec K, and a com- bination of (P1) and (P4) would imply the inclusion SL ⊂ lin K, contrary to the first assertion. We will need one more technical lemma. Lemma 3.2. Let K ⊂ Rn be a unbounded closed convex set, and u ∈ K, v ∈ Rn be points. Any unbounded sequence of points x1,x2, . . . ∈ K \{u,v} contains a subsequence, say x′1,x ′ 2, . . . , such that the unit vectors ei = x′i −u ‖x′i −u‖ and ci = x′i −v ‖x′i −v‖ , i ≥ 1, (1) converge to the same unit vector e ∈ rec K. Proof. Since all vectors (xi−u)/‖xi−u‖, i ≥ 1, belong to the unit sphere of Rn, a compactness argument implies the existence of a subsequence, say x′1,x ′ 2, . . . , of x1,x2, . . . such that the vectors e1,e2, . . . given by (1) converge to a unit vector e. The inequalities ‖x′i‖−‖u‖≤‖x ′ i −u‖≤‖x ′ i‖ + ‖u‖, ‖x′i‖−‖v‖≤‖x ′ i −v‖≤‖x ′ i‖ + ‖v‖ cone asymptotes of convex sets 87 give lim i→∞ ‖x′i −u‖ = lim i→∞ ‖x′i −v‖ = ∞ and lim i→∞ ‖x′i −u‖ ‖x′i −v‖ = 1. Therefore, lim i→∞ ci = lim i→∞ ( x′i −u ‖x′i −v‖ + u−v ‖x′i −v‖ ) = lim i→∞ ( ‖x′i −u‖ ‖x′i −v‖ x′i −u ‖x′i −u‖ + u−v ‖x′i −v‖ ) = 1e + o = e. Next, we assert that the closed halfline [u,u + e〉 lies in K. Indeed, choose any point y in [u,u + e〉. Then y = (1 −λ)u + λ(u + e) = u + λe, λ ≥ 0. Clearly, ‖y − u‖ = λ. Choose an integer i0 such that ‖x′i − u‖ ≥ max{1,λ} for all i ≥ i0. Then u + ei = ( 1 − 1 ‖x′i −u‖ ) u + 1 ‖x′i −u‖ x′i ∈ [u,x ′ i], i ≥ i0. With yi = u + λei, one has yi = (1 −λ)u + λ(u + ei) ∈ [u,u + ei〉 = [u,x′i〉. Now, the inequality ‖yi −u‖ = λ ≤‖x′i −u‖, i ≥ i0, gives yi ∈ [u,x′i] ⊂ K for all i ≥ i0. Finally, y = u + λe = lim i→∞ (u + λei) = lim i→∞ yi ∈ K. Summing up, [u,u + e〉⊂ K. Finally, (P2) shows that e ∈ rec K. Theorem 3.3. Let K ⊂ Rn be a closed convex set, and let L ⊂ Rn be a plane asymptote of K. With C = rec K ∩SL, the following assertions hold. (a) For any point v ∈ L and a scalar ε > 0, the closed cone v + Dε(C)∩SL lies in L and is an asymptote of K. Furthermore, there is a scalar ε0 > 0 such that v + Dε(C) ∩SL is a proper subset of L for all 0 < ε < ε0. 88 v. soltan (b) If a closed subset M of L is an asymptote of K, then, for any point v ∈ L and a scalar ε > 0, the closed set M ∩ (v + Dε(C) ∩ SL) is an asymptote of K. Proof. (a) Since rec K is a closed convex cone with apex o, so is the set C. By Lemma 3.1, {o} 6= C 6= SL. Then the polar cone C◦L of C within the space SL is nontrivial. If e is a nonzero vector in C ◦ L, then C lies in the closed halfplane P = {x ∈ SL : x·e ≤ 0} of SL. A simple geometric argument shows the existence of a scalar ε0 > 0 such that the closed cone Dε(C) ∩ SL is a proper subset of SL for all 0 < ε < ε0. Obviously, v + Dε(C) ∩SL is a closed cone with apex v. Furthermore, v + Dε(C) ∩SL ⊂ v + SL = L. By the above argument, v + Dε(C) ∩ SL is a proper subset of L for all 0 < ε < ε0. Because L is an asymptote of K, there are sequences of points p1,p2, · · · ∈ K and q1,q2, . . . ∈ L such that limi→∞‖pi − qi‖ = 0. Clearly, both sets {p1,p2, . . .} and {q1,q2, . . .} are unbounded. Indeed, otherwise, one could choose respective subsequences converging to the same point in K∩L, contrary to the assumption K ∩L = ∅. Choose a point u ∈ K. By Lemma 3.2, there is an unbounded subsequence pi1,pi2, . . . such that the unit vectors eij = pij −u ‖pij −u‖ and cij = pij −v ‖pij −v‖ , j ≥ 1, converge to the same unit vector e ∈ rec K. Let c′ij = qij −v ‖qij −v‖ , j ≥ 1. Since the subsequence qi1,qi2, . . . is unbounded and ‖pij −qij‖→ 0 as j →∞, the inequalities ‖qij −v‖−‖pij −qij‖≤‖pij −v‖≤‖qij −v‖ + ‖pij −qij‖ give lim j→∞ ‖pij −v‖ ‖qij −v‖ = 1. cone asymptotes of convex sets 89 Therefore, lim j→∞ c′ij = limj→∞ ( pij −v ‖qij −v‖ − pij −qij ‖qij −v‖ ) = lim j→∞ ( ‖pij −v‖ ‖qij −v‖ pij −v ‖pij −v‖ − pij −qij ‖qij −v‖ ) = 1e + o = e. Because c′ij ∈ SL for all j ≥ 1, the limit vector e also belongs to SL. Thus e ∈ rec K ∩SL = C. Clearly, there is an index r ≥ 1 such that every vector eij , j ≥ r, forms with e an angle of size at most ε. Equivalently, every halfline [o,qij − v〉 forms with the halfline h = [o,e〉 ⊂ C an angle of size at most ε. Thus qij ∈ [v,qij〉 = v + [o,qij −v〉⊂ v + Dε(h) ⊂ v + Dε(C) ∩SL, j ≥ r. Finally, because limj→∞‖pij − qij‖ = 0, the cone v + Dε(C) ∩ SL is an asymptote of K. (b) Suppose that a closed subset M of L is an asymptote of K. Then there are sequences of points p1,p2, . . . ∈ K and q1,q2, . . . ∈ M such that limi→∞‖pi−qi‖ = 0. An obvious modification of the argument from part (a) shows the existence of unbounded subsequences pi1,pi2, . . . ∈ K and qi1,qi2, . . . ∈ M ∩ (v + Dε(C) ∩SL) such that limj→∞‖pij − qij‖ = 0. So, the set M ∩ (v + Dε(C) ∩ SL) is an asymptote of K. We observe that, generally, Theorem 2.1 does not follow from Theorem 3.3. Indeed, if K is the convex set from Example 2.2, then its recession cone is the polyhedron {(x,y,z) : x ≥ 0, y ≥ 0}. In terms of Theorem 3.3, SL = L, C is the halfplane {(x, 0,z) : x ≥ 0} of L, and v + Dε(C)∩SL is the nonconvex closed cone in L of angle size π + 2ε. Clearly, this cone is larger than any halfplane asymptote of K contained in v + Dε(C)∩SL, and there is no halfplane asymptote of K containing this cone. Analysis of the proof of Theorem 3.3 results in the following corollary, which shows the existence (but not a constructive description) of arbitrarily sharp pencil cone asymptotes. Corollary 3.4. Let K ⊂ Rn be a closed convex set, and L ⊂ Rn be a plane asymptote of K. There is a closed halfline h ⊂ rec K ∩ SL with endpoint o such that for any point v ∈ L and scalar ε > 0, the pencil cone (v + Dε(h)) ∩L is an asymptote of K. 90 v. soltan The next example shows that in Theorem 3.3, the set Dε(C) cannot be replaced with C and even with the closed metric ε-neighborhood of C, defined by Nε(C) = {x ∈ Rn : ρ(x,C) ≤ ε}. Consequently, no translate of the cone v + C may be an asymptote of K. Example 3.5. The set K ⊂ R3 given by the conditions K = {(x,y,z) : x > 0, y ≥ 1/x, z ≥ (x + y)2} is closed and convex. Furthermore, the following properties of K hold: (a) the coordinate xz-plane, say L, is an asymptote of K; (b) no closed slab between a pair of parallel lines in L is an asymptote of K; (c) rec K is the vertical halfline h = {(0, 0,z) : z ≥ 0}, and no metric neighborhood v + Nε(h) ∩ L of the halfline v + h, with v ∈ L, is an asymptote of K; (d) for any choice of scalars α ≥ 0, and β,γ ∈ R, the planar cone D(α,β,γ) = {(x, 0,z) : z ≥ α|x−β| + γ}⊂ L (2) is an asymptote of K. Indeed, the set K is closed and convex as the intersection of closed and convex sets K1 = {(x,y,z) : x > 0, y ≥ 1/x} and K2 = {(x,y,z) : z ≥ (x + y)2}. (a) Clearly, K ∩L = ∅, and the sequences of points pi = ( i, 1/i, [(i2 + 1)/i]2 ) ∈ K, qi = ( i, 0, [(i2 + 1)/i]2 ) ∈ L, i ≥ 1, satisfy the condition ‖pi − qi‖ = 1/i. So, δ(K,L) = 0, which shows that L is an asymptote of K. (b) Let Q be any closed slab between a pair of parallel lines in L. Assume first that Q is vertical. Then it can be described as Q = {(x, 0,z) : α ≤ x ≤ β}, where α ≤ β. cone asymptotes of convex sets 91 Since K1∩Q = ∅ and K1 is a both-way unbounded vertical cylinder, δ(K1,Q) equals the distance between the branch of hyperbola {(x,y, 0) : x > 0,xy = 1} ⊂ bd K1 and the point (α, 0, 0) ∈ Q, which is positive. Therefore, the inclusion K ⊂ K1 gives δ(K,Q) ≥ δ(K1,Q) > 0. Hence no vertical slab in L is an asymptote of K. Suppose now that Q is slant. Then Q = {(x, 0,z) : αx + β ≤ z ≤ αx + β′} for suitable scalars α and β ≤ β′. Assume for a moment that δ(K,Q) = 0. Then one can choose sequences of points pi ∈ K and qi ∈ Q, i ≥ 1, such that ‖pi − qi‖→ 0 as i →∞. Denote by p′i the point at which the segment [pi,qi] meets bd K, i ≥ 1. Let p′i = (xi,yi,zi) and qi = (ui, 0,vi), αui + β ≤ vi ≤ αui + β ′, i ≥ 1. By a convexity argument, xiyi = 1 and ‖p′i−qi‖≤‖pi−qi‖. Hence ‖p ′ i−qi‖→ 0 as i →∞, or, equivalently, lim i→∞ ( (xi −ui)2 + 1/x2i + (zi −vi) 2 ) = 0. (3) Thus xi →∞ and ui = xi + εi, where εi → 0 as i →∞. Consequently, there is an index i0 ≥ 0 and a scalar γ > 0 such that |vi| ≤ max { |αui + β|, |αui + β′| } ≤ |αxi| + |αεi| + max{|β|, |β′|}≤ |αxi| + γ whenever i ≥ i0. Therefore, |zi −vi| ≥ |zi|− |vi| ≥ ( (xi + 1/xi) 2 −|αxi|−γ ) →∞ as i →∞, in contradiction with (3). Summing up, Q cannot be an asymptote of K. (c) Given the point p = (1, 1, 4) ∈ K, the halfline p+h = {(1, 1,z) : z ≥ 4} is included in K. This argument and (P2) imply that h ⊂ rec K. For the opposite inclusion, choose any vector e = (u,v,w) ∈ rec K. Then p + λe = (1 + λu, 1 + λv, 4 + λw) ∈ K for all λ > 0. Equivalently, for all λ > 0, one has 1 + λu > 0, 1 + λv ≥ 1/(1 + λu), 4 + λw ≥ (2 + λ(u + v))2. 92 v. soltan The first condition implies that u ≥ 0. This inequality and the second condi- tion give v ≥ 0, while the third condition results in u+v = 0. Hence u = v = 0 and w ≥ 0, implying the inclusion e ∈ h. Summing up, rec K = h. It is easy to see that a metric neighborhood v + Nε(h) ∩ L of v + h, with v ∈ L, is a subset of a suitable vertical slab Q ⊂ L. According to part (b) above, Q is not an asymptote of K. Hence v + Nε(h) ∩L is not an asymptote of K. (d) Let SL = L and v = (β, 0,γ). If α > 0, then the planar cone D(α,β,γ) from (2) can be expressed as v + Dε(h) ∩SL, where ε = cot−1(α). If α = 0, then D(0,β,γ) = v + Dπ/2(h) ∩SL. In either case, Corollary 3.4 shows that D(α,β,γ) is an asymptote of K. 4. Line-free cone asymptotes We describe below a family of plane asymptotes M contained in a plane asymptote L of K such that the cone Dε(rec K ∩M) ∩SM becomes line-free provided ε is sufficiently small. Theorem 4.1. Let K ⊂ Rn be a closed convex set, L ⊂ Rn be a plane asymptote of K, and F be a subspace complementary to lin K ∩ SL in SL. The following assertions hold. (a) For any point v ∈ L, the plane M = v+F lies in L and is an asymptote of K. (b) For any point v ∈ L and any ε > 0, the closed cone v+Dε(rec K∩F)∩F is a line-free asymptote of K. (c) There is an n-dimensional line-free closed convex cone C′ ⊂ Rn with apex o such that C′ ⊂ Dε(rec K ∩ F) and the cone v + C′ ∩ F is an asymptote of K. The proof of Theorem 4.1 is divided into Lemmas 4.2 – 4.4. Lemma 4.2. Let K ⊂ Rn be a closed convex set, L ⊂ Rn be a plane asymptote of K, and v be a point in L. For any subspace F ⊂ Rn com- plementary to lin K ∩ SL in SL, the plane M = v + F lies in L and is an asymptote of K. Proof. Since lin K ⊂ rec K, Lemma 3.1 shows that the subspace T = lin K ∩ SL is distinct from SL (possibly, T = {o}). By (P3), K = K + T. cone asymptotes of convex sets 93 Choose a subspace F ⊂ Rn complementary to T in SL and consider the plane M = v + F. Since M = v + F ⊂ v + SL = L and K ∩ L = ∅, it remains to show that δ(K,M) = 0. Indeed, choose sequences of points p1,p2, . . . ∈ K and q1,q2, . . . ∈ L such that limi→∞‖pi − qi‖ = 0. Clearly, qi − v ∈ SL, i ≥ 1. Denote by q′i the projection of qi − v on F in the direction of T, and let p′i = pi + (v + q ′ i −qi), i ≥ 1. Because v + q′i −qi = q ′ i − (qi −v) ∈ T, one has p′i = pi + (v + q ′ i −qi) ∈ K + T = K, i ≥ 1. Furthermore, v + q′i ∈ v + F = M, i ≥ 1, and δ(K,M) ≤ lim i→∞ ‖p′i − (v + q ′ i)‖ = lim i→∞ ‖(pi + v + q′i −qi) − (v + q ′ i)‖ = lim i→∞ ‖pi −qi‖ = 0. Summing up, M is an asymptote of K. Lemma 4.3. Let K ⊂ Rn be a closed convex set, and L ⊂ Rn be a plane asymptote of K. Given a subspace F complementary to lin K∩SL in SL, the closed convex cone C = rec K ∩F with apex o is nontrivial and line-free. Proof. By Lemma 4.2, every translate of the form v+F, where v ∈ L, is an asymptote of K. Consequently, Lemma 3.1 shows that the closed convex cone C = rec K ∩ F is distinct from {o}. It remains to prove that C is line-free. Assume for a moment that C contains a line l. We consider separately the following two cases. (i) Let o ∈ l. By (P4), l is a 1-dimensional subspace contained in lin K. Consequently, l ⊂ lin K ∩F, contrary to the choice of F. (ii) Let o /∈ l. Because o is an apex of rec K, every closed halfline [o,x〉, where x ∈ l, is contained in rec K. Clearly, the union of all such halflines coincides with {o}∪ P , where P is the open halfplane of the 2-dimensional subspace, span l, bounded by the 1-dimensional subspace l′ = l − l. Then l′ ⊂ cl P ⊂ rec K, and, as above, l′ ∈ lin K ∩F, contrary to the choice of F . Summing up, C should be line-free. 94 v. soltan Lemma 4.4. If C ⊂ Rn is a closed line-free convex cone with apex o, then there is a scalar ε > 0 such that the closed cone Dε(C) is line-free. Furthermore, there is an n-dimensional closed convex cone C′ ⊂ Rn with apex o and a scalar ε′ ∈ (0,ε) such that Dε′(C) ⊂ C′ ⊂ Dε(C). Proof. Since C is line-free, the polar cone C◦ is n-dimensional (see, e.g., [10, Theorem 8.4]). Choose a nonzero vector e ∈ (− int C◦). Then C \{o} is contained in the open halfspace W = {x ∈ Rn : x·e > 0} (see [10, Theorem 8.6]). Denote by S the unit sphere of Rn, and let E = C ∩ S. Clearly, C = {λx : λ ≥ 0, x ∈ E}. Since E is a compact subset of W , there is an ε > 0 such that the closed set Vρ(E) = {x ∈ S : δ(x,E) ≤ ρ}, ρ = 2 sin(ε/2), is contained in W . We observe that a closed halfline h = [o,u〉, with u ∈ S, is contained in Dε(C) if and only if there is a closed halfline h ′ = [o,u′〉, with u′ ∈ E, such that the angle between h and h′ is of size at most ε. Considering the isosceles triangle ∆(o,u,u′), we deduce that the angle between h and h′ is of size at most ε if and only if ‖u−u′‖≤ 2 sin(ε/2). This argument implies that Dε(C) = {λx : λ ≥ 0, x ∈ Vρ(E)}. It is easy to see that every closed halfline [o,x〉, where x ∈ Vρ(E), is contained in {o}∪W . Hence Dε(C) ⊂{o}∪W . An argument similar to that of Lemma 4.3 shows that the cone Dε(C) is line-free. Finally, consider the hyperplane H = {x ∈ Rn : x·e = 1}. Then both sets A = C ∩ H and B = Dε(C) ∩ H are compact (see [10, Theorem 8.15]), and A is convex. Furthermore, both sets A and B have dimension n− 1 and rint A ⊂ rint B (see [10, Theorem 8.14]). Given any point a ∈ rint A, the set A′ = a+µ(A−a), with µ > 1, is convex and contains A in its relative interior. A compactness argument implies that A′ ⊂ B provided µ is sufficiently close to 1. For this value of µ, let C′ = Co(A ′) = {λx : λ ≥ 0, x ∈ A′}. Then C′ is a closed convex cone satisfying the inclusions C ⊂{o}∪ int C′ and C′ ⊂ Dε(C), as desired. The following example shows that the cone Dε(C) in Lemma 4.4 may be nonconvex even if the cone C is convex and line-free. cone asymptotes of convex sets 95 Example 4.5. Let C = {(x,y, 0) : y ≥ |x|/10} be the planar cone in R3. Clearly, C is closed, convex, and line-free. Let ε = sin−1(0.1) ≈ 0.1. The point u = (10, 1, 1) belongs to Dε(C) because u ′ = (10, 1, 0) ∈ C and the angle between the halflines [o,u〉 and [o,u′〉 equals sin−1(1/ √ 102) < ε. Similarly, the point v = (−10, 1, 1) belongs to Dε(C). On the other hand, the midpoint w = (0, 1, 1) of the segment [u,v] does not belong to Dε(C). Indeed, the angle between [o,w〉 and C is achieved on the pair of halflines [o,w〉 and [o,w′〉, where w = (0, 1, 0), and this angle equals sin−1(1/ √ 2) = π/4 > ε. So, w /∈ Dε(C), implying that the cone Dε(C) is not convex. In view of Theorem 4.1, one may ask about the existence of minimal (under inclusion) cone asymptotes of K. The following theorem shows that such asymptotes (if any) should be halflines. Theorem 4.6. Let X ⊂ Rn be a nonempty closed set, and C ⊂ Rn be a closed convex cone with apex v, which is an asymptote of X. If dim C ≥ 2, then there is a convex cone C′ with apex v which is an asymptote of K and a proper subset of C. Proof. Choose a point u ∈ rint C\{v}, and consider a hyperplane H ⊂ Rn containing {u,v} such that C meets the interiors of both closed halfspaces, say V1 and V2, determined by H. The sets C1 = C ∩V1 and C2 = C ∩V2 are closed convex cones with common apex v, whose union is C, and each of these cones is a proper subset of C. Because C is an asymptote of K, there is an unbounded sequence of points x1,x2, . . . ∈ C such that δ(xi,X) → 0 as i →∞. Clearly, one of the cones C1 and C2, say C1, contains an infinite subsequence x′1,x ′ 2, . . . of x1,x2, . . . . Since C1 ∩ X ⊂ C ∩ X = ∅ and δ(x ′ i,X) → 0 as i →∞, the cone C1 is an asymptote of X. 5. Properties of cone asymptotes The next two theorems show that some properties of plane asymptotes can be generalized to the case of cone asymptotes (see [4] and [9] for the original statements). Theorem 5.1. Let C ⊂ Rn be a closed convex cone with apex o. If a translate of C is an asymptote of a closed convex set K ⊂ Rn, then rec K ∩C 6= {o}. Furthermore, for any nonzero vector e ∈ rec K ∩C, there is a translate of the closed halfline h = [o,e〉 which either lies in bd K or is an asymptote of K. 96 v. soltan Proof. Let a translate D = v + C of C be an asymptote of K. Then K ∩D = ∅ and there are sequences of points p1,p2, . . . ∈ K and q1,q2, . . . ∈ D such that limi→∞‖pi − qi‖ = 0. We observe that both sequences are un- bounded. Indeed, otherwise one could choose respective subsequences con- verging to the same point in K ∩D, contrary to the assumption K ∩D = ∅. As shown in the proof of Theorem 3.3, there are subsequences pi1,pi2, . . . and qi1,qi2, . . . such that the unit vectors eij = pij −u ‖pij −u‖ an d c′ij = qij −v ‖qij −v‖ , j ≥ 1, converge to the same unit vector e ∈ rec K. Since qij − v ∈ C for all j ≥ 1 and the cone C is closed, we obtain that e ∈ C. Consider the halfline h = [o,e〉. Then h ⊂ C and, by (P2), the halfline u + h lies in K. Consider the family of parallel halflines h(λ) = ((1 −λ)v + λu) + h, 0 ≤ λ ≤ 1. By the above, h(0) ⊂ D and h(1) ⊂ K. Consequently, K ∩h(0) = ∅. Let λ′ = sup{λ ∈ [0, 1] : K ∩h(λ) = ∅}. It is easy to see that int K∩h(λ′) = ∅ and δ(K,h(λ′)) = 0. If K∩h(λ′) = ∅, then h(λ′) is an asymptote of K. Suppose that K ∩h(λ′) 6= ∅ and choose a point w ∈ K ∩h(λ). By (P2), the halfline w + h lies in K, and the inclusion w + h ⊂ h(λ′) implies that w + h ⊂ bd K, as desired. The following example shows that in Theorem 5.1 (unlike Lemma 3.1) the inclusion rec K ∩C ⊂ lin K is possible. Example 5.2. Let the closed convex sets K ⊂ R3 be given by K = {(x,y,z) : x ≥ 0, y ≥ 0, x + y ≥ 1}. Clearly, rec K = {(x,y,z) : x ≥ 0, y ≥ 0} and lin K is the z-axis. Let C be the closed convex cone with apex o generated by the circular disk D = {(x,y, 1) : x2 + (y + 1)2 ≤ 1}. The cone C is an asymptote of K because K ∩ C = ∅ and δ(C,l) = 0, where l ⊂ K is the vertical line {(1, 0,z) : z ∈ R}. At the same time, rec K ∩C = {(0, 0,z) : z ≥ 0} is a subset of lin K. cone asymptotes of convex sets 97 Theorem 5.3. Let X ⊂ Rn be a nonempty closed set and C ⊂ Rn be a closed convex cone with apex o. There is a translate of C which is an asymptote of X if and only if the set X −C is not closed. Proof. Let a translate D = v + C of C be an asymptote of X. We first observe that v /∈ X−C. Indeed, assume for a moment that v ∈ X−C. Then v = x−u, where x ∈ X and u ∈ C. Consequently, x = v + u ∈ v + C = D, contrary to the assumption D ∩ X = ∅. Since D is an asymptote of X, there are sequences of points x1,x2, . . . ∈ X and y1,y2, . . . ∈ D such that limi→∞‖xi−yi‖ = 0. Expressing yi as yi = v + zi for a suitable zi ∈ C, i ≥ 1, one has xi −zi ∈ X −C and lim i→∞ ‖v − (xi −zi)‖ = lim i→∞ ‖yi −xi‖ = 0. So, v ∈ cl(X −C) \ (X −C). Thus the set X −C is not closed. Conversely, suppose that the set X −C is not closed. Choose a point v ∈ cl(X −C) \ (X −C) and consider the cone D = v +C. We assert that D∩X = ∅. Indeed, assume the existence of a point z ∈ D ∩X. Then z = v + u = x, where u ∈ C and x ∈ X. Consequently, v = x−u ∈ X −C, which is impossible by the choice of v. Next, the inclusion v ∈ cl(X − C) implies the existence of a sequence u1,u2, . . . of points in X − C converging to v. With ui = xi − zi, where xi ∈ X and zi ∈ C, and with wi = v + zi ∈ D, i ≥ 1, we obtain that lim i→∞ ‖xi −wi‖ = lim i→∞ ‖(ui + zi) − (v + zi)‖ = lim i→∞ ‖ui −v‖ = 0. Hence δ(X,D) = 0. Summing up, D is an asymptote of X. Acknowledgements The author is thankful to the referee for the careful reading and considered suggestions leading to a better presented paper. 98 v. soltan References [1] A. Auslender, M. Teboulle, “Asymptotic Cones and Functions in Opti- mization and Variational Inequalities”, Springer-Verlag, New York, 2003. [2] D. Gale, V. Klee, Continuous convex sets, Math. Scand. 7 (1959), 379 – 391. [3] P. Goossens, Hyperbolic sets and asymptotes, J. Math. Anal. 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Soltan, On M-decomposable sets, J. Math. Anal. Appl. 485 (2020), 123816, 15 pp. Introduction Halfplane asymptotes Cone asymptotes Line-free cone asymptotes Properties of cone asymptotes