A "hidden" characterization of approximatively polyhedral convex sets in Banach spaces
aa r X i v : . [ m a t h . F A ] N ov A “HIDDEN” CHARACTERIZATION OFAPPROXIMATIVELY POLYHEDRAL CONVEX SETS IN BANACH SPACES
TARAS BANAKH AND IVAN HETMAN
Abstract.
For a Banach space X by Conv H ( X ) we denote the space of non-empty closed convex subsets of X ,endowed with the Hausdorff metric. We prove that for any closed convex set C ⊂ X and its metric component H C = { A ∈ Conv H ( X ) : d H ( A, C ) < ∞} in Conv H ( X ), the following conditions are equivalent:(1) C is approximatively polyhedral, which means that for every ε > P ⊂ X on Hausdorff distance d H ( P, C ) < ε from C ;(2) C lies on finite Hausdorff distance d H ( C, P ) from some polyhedral convex set P ⊂ X ;(3) the metric space ( H C , d H ) is separable;(4) H C has density dens( H C ) < c ;(5) H C does not contain a positively hiding convex set P ⊂ X .If the Banach space X is finite-dimensional, then the conditions (1)–(5) are equivalent to:(6) C is not positively hiding;(7) C is not infinitely hiding.A convex subset C ⊂ X is called positively hiding (resp. infinitely hiding ) if there is an infinite set A ⊂ X \ C suchthat inf a ∈ A d ( a, C ) > a ∈ A d ( a, C ) = ∞ ) and for any distinct points a, b ∈ A the segment [ a, b ] meets the set C . Introduction
In [1] the authors proved that a closed convex subset C of a complete linear metric space X is polyhedral in itslinear hull if and only if C hides no infinite subset A ⊂ X \ C in the sense that [ a, b ] ∩ C = ∅ for any distinct points a, b ∈ A . In this paper we shall prove a similar “hidden” characterization of approximatively polyhedral subsetsin Banach spaces, simultaneously giving a characterization of separable components of the space of closed convexsubsets Conv H ( X ) of a Banach space X .For a Banach space X by Conv H ( X ) we denote the space of all non-empty closed convex subsets of X , endowedwith the Hausdorff metric d H ( A, B ) = max { sup a ∈ A dist ( a, B ) , sup b ∈ B dist ( b, A ) } ∈ [0 , ∞ ] . Here dist ( a, B ) = inf b ∈ B k a − b k stands for the distance from a point a ∈ X to a subset B ⊂ X of the Banach space X .It is well-known that for each closed convex set C ∈ Conv H ( X ) the Hausdorff distance d H restricted to the set H C = { A ∈ Conv H ( X ) : d H ( A, C ) < ∞} is a metric, see [9, Ch2]. The obtained metric space ( H C , d H ) will be called the Hausdorff metric component (orjust component ) of the set C in Conv H ( X ).In fact, the present investigation was motivated by the problem of calculating the density of components of thespace Conv H ( X ) and detecting closed convex subsets C ⊂ X with separable component H C . In this paper we shallcharacterize such sets C in terms of approximative polyhedrality as well as in “hidden” terms resembling those from[1].A convex subset C of a Banach space X is called • a closed half-space if C = f − ([ a, + ∞ ) for some non-zero linear continuous functional f : X → R and somereal number a ∈ R ; • polyhedral if C can be written as the intersection C = ∩F of a finite family F of closed half-spaces in X ; • approximatively polyhedral if for every ε > P ⊂ X that lies on Hausdorffdistance d H ( C, P ) < ε from C . Mathematics Subject Classification.
Key words and phrases.
Polyhedral convex set; approximatively polyhedral convex set, positively hiding convex set; infinitely hidingconvex set, space of closed convex sets, Hausdorff metric.
Observe that the whole space X is polyhedral, being the intersection X = ∩F of the empty family F = ∅ ofclosed half-spaces.It is well-known that each compact convex subset of Banach space is approximatively polyhedral (see [6] formore information on that topic). This is not necessarily true for non-compact closed convex sets. For example, theconvex parabola P = (cid:8) ( x, y ) ∈ R : y ≥ x (cid:9) is not approximatively polyhedral in R , while the convex hyperbola H = (cid:8) ( x, y ) ∈ R : y ≥ p x + 1 (cid:9) is approximatively polyhedral.Next, we introduce some “hidden” properties of convex sets. Following [1], we say that a subset C of a linearspace X hides a set A ⊂ X if for any two distinct points a, b ∈ A the affine segment [ a, b ] = { ta + (1 − t ) b : t ∈ [0 , } meets the set C .A convex subset C of a Banach space X is called • hiding if C hides some infinite set A ⊂ X \ C ; • positively hiding if C hides some infinite set A ⊂ X \ C such that inf a ∈ A dist ( a, C ) > • infinitely hiding if C hides some infinite set A ⊂ X \ C such that sup a ∈ A dist ( a, C ) = ∞ .It is clear that each infinitely hiding set is positively hiding and each positively hiding set is hiding.By [1], a closed convex subset C of a complete linear metric space X is hiding if and only if C is not poly-hedral in its closed linear hull. So, both parabola and hyperbola are hiding (being not polyhedral). Yet, theparabola is infinitely hiding (but not approximatively polyhedral) while the hyperbola is not positively hiding (butis approximatively polyhedral).It turns out that the approximative polyhedrality and positive or infinite hiding properties are mutually exclusive,and can be characterized via properties of the characteristic cone of a given convex set.Let us recall that the characteristic cone of a convex subset C in a linear topological space X is the set V C ofall vectors v ∈ X such that for every point c ∈ C the ray c + ¯ R + v = { c + tv : t ≥ } lies in C . Here ¯ R + = [0 , ∞ )denotes the closed half-line. The characteristic cone V C is closed in X if C is closed or open in X , see Lemma 2.2.The main result of this paper is the following characterization theorem that will be used in the paper [2] devotedto recognizing the topological structure of the space Conv H ( X ). In finite-dimensional case, the equivalence of theconditions (1)–(3) was proved by Viktor Klee in [8]. Theorem 1.1.
For a closed convex subset C of a Banach space X the following conditions are equivalent: (1) C is approximatively polyhedral; (2) the characteristic cone V C of C is polyhedral in X and d H ( C, V C ) < ∞ ; (3) the component H C contains a polyhedral closed convex set; (4) the component H C contains no positively hiding closed convex set; (5) the space H C is separable; (6) the space H C has density dens( H C ) < c .If the Banach space X is finite-dimensional, then the conditions (1)–(6) are equivalent to: (7) C is not positively hiding; (8) C is not infinitely hiding. Let us recall that the density dens( X ) of a topological space X is the smallest cardinality | D | of a dense subset D of X . Topological spaces with at most countable density are called separable . Remark 1.
Observe that the closed unit ball C = { x ∈ l : k x k ≤ } in the separable Hilbert space l ispositively hiding but not infinitely hiding. This example shows that the conditions (7) and (8) are not equivalentin infinite-dimensional Banach spaces.Theorem 1.1 will be proved in Section 7 after long preliminary work made in Sections 2–6.2. Some properties of characteristic cones
This section is of preliminary character and contains some information on convex cones in Banach spaces. Alllinear (and Banach) spaces considered in this paper are over the field of real numbers R .By a convex cone in a linear space X we understand a convex subset C ⊂ X such that tc ∈ C for any t ∈ ¯ R + and c ∈ C . Here ¯ R + = [0 , + ∞ ) stands for the closure of the open half-line R + = (0 , + ∞ ) in R . For two subsets N APPROXIMATIVELY POLYHEDRAL CONVEX SETS 3
A, B of a linear space X and a real number λ , let A + B = { a + b : a ∈ A, b ∈ B } be the pointwise sum of A and B , and λA = { λ · a : a ∈ A } be a homothetic copy of A .Each subset F ⊂ X generates the conecone( F ) = n n X i =1 λ i x i : n ∈ N and ( x i ) ni =1 ∈ F n , ( λ i ) ni =1 ∈ ¯ R n + o , which contains the convex hull conv( F ) of F .The following description of polyhedral cones and polyhedral convex sets in finite-dimensional spaces is classicaland can be found in [8], [10, Theorems 1.2, 1.3] or [4, § Lemma 2.1.
Let X be a finite-dimensional Banach space. (1) A convex cone C ⊂ X is polyhedral if and only if C = cone( F ) for some finite set F ⊂ X . (2) A convex set C ⊂ X is polyhedral if and only if C = cone( F ) + conv( E ) for some finite sets F, E ⊂ X . We shall be mainly interested in characteristic cones and dual characteristic cones of convex sets in Banachspaces. Let us recall that for a convex subset C of a Banach space X its characteristic cone V C is defined by V C = { x ∈ X : ∀ c ∈ C c + ¯ R + x ⊂ C } ⊂ X. By the dual characteristic cone of C we understand the convex cone V ∗ C = { x ∗ ∈ X ∗ : sup x ∗ ( C ) < ∞} lying in the dual Banach space X ∗ .It is clear that the dual characteristic cone V ∗ C of a convex set C ⊂ X coincides with the dual characteristic cone V ∗ ¯ C of its closure ¯ C in X . The relation between the characteristic cones V C and V ¯ C are described in the followingsimple lemma, whose proof is left to the reader as an exercise. Lemma 2.2.
Let ¯ C be the closure of a convex set C in a Banach space X . Then (1) V C ⊂ V ¯ C ; (2) V C = V ¯ C if the set C is open in X . Our next aim is to show that the characteristic cones of two closed convex subsets
A, B ⊂ X with d H ( A, B ) < ∞ coincide. For this we shall need: Lemma 2.3.
For each point c of a convex set C in a Banach space X , each vector v / ∈ V ¯ C and each real number ε ∈ R + there is a real number t ∈ R + such that dist ( c + tv, C ) = ε .Proof. Since v / ∈ V ¯ C , there is a real number t > c + t v / ∈ ¯ C . Consider the continuous function f : ¯ R + → ¯ R + , f : t dist ( c + tv, C ) , and observe that f (0) = 0 as c ∈ C . We claim that lim t → + ∞ f ( t ) = + ∞ . Since c + t v / ∈ ¯ C , we can apply Hahn-Banach Theorem and find a linear functional x ∗ ∈ X ∗ with unit norm such that x ∗ ( c + t v ) > sup x ∗ ( ¯ C ) ≥ x ∗ ( c ),which implies that x ∗ ( v ) >
0. Then for any real number t > t we get dist ( tv, C ) = inf c ∈ C k c + tv − c k ≥ inf c ∈ C | x ∗ ( tv ) − x ∗ ( c − c ) | = tx ∗ ( v ) − sup x ∗ ( C − c )and hence lim t → + ∞ dist ( c + tv, C ) = ∞ . By the continuity of f , there is a positive real number t with dist ( c + tv, C ) = f ( t ) = ε . (cid:3) Now we can prove the promised:
Lemma 2.4.
Let
A, B be two closed convex sets in a Banach space X . If d H ( A, B ) < ∞ , then V A = V B .Proof. We lose no generality assuming that 0 ∈ A ∩ B . If V A = V B , then we can find a vector v ∈ X that lies in V A \ V B or in V B \ V A . We lose no generality assuming that v ∈ V B \ V A . By Lemma 2.3, there is a positive realnumber t such that dist ( tv, A ) > d H ( A, B ), which is not possible as tv ∈ V B ⊂ B . (cid:3) Observe that for a convex C containing zero, the inclusion V C ⊂ C implies V ∗ C ⊂ V ∗ V C . Lemma 2.5.
For any closed convex set C in a Banach space the dual characteristic cone V ∗ V C of the characteristiccone V C of C coincides with the weak ∗ closure cl ∗ ( V C ) of V C . TARAS BANAKH AND IVAN HETMAN
Proof.
We lose no generality assuming that 0 ∈ C . Observe that the dual characteristic cone V ∗ V C = { x ∗ ∈ X ∗ : sup x ∗ ( V C ) < ∞} = { x ∗ ∈ X ∗ : sup x ∗ ( V C ) = 0 } = \ v ∈ V C { x ∗ ∈ X ∗ : x ∗ ( v ) ≤ } is weak ∗ closed in X ∗ , being an intersection of weak ∗ -closed half-spaces in X ∗ . So, the inclusion V ∗ C ⊂ V ∗ V C impliescl ∗ ( V ∗ C ) ⊂ V ∗ V C . It remains to prove the reverse inclusion V ∗ V C ⊂ cl ∗ ( V ∗ C ). Assume conversely that it is not true andfind a functional x ∗ ∈ V ∗ V C \ cl ∗ ( V ∗ C ). By the Hahn-Banach Theorem applied to the weak ∗ topology of X ∗ , there isan element x ∈ X that separates x ∗ from cl ∗ ( V ∗ C ) in the sense that x ∗ ( x ) > sup { v ∗ ( x ) : v ∗ ∈ cl ∗ ( V ∗ C ) } ≥ sup { v ∗ ( x ) : v ∗ ∈ V ∗ C } . We claim that v ∗ ( x ) ≤ v ∗ ∈ V ∗ C . Assuming that v ∗ ( x ) >
0, we can find a positive real number λ so large that λv ∗ ( x ) > x ∗ ( x ), which contradicts the choice of x (because λv ∗ ∈ V ∗ C ). So, v ∗ ( x ) ≤ v ∗ ∈ V ∗ C .We claim that x ∈ V C .In the opposite case, we could find a positive real number t > tx / ∈ C (here we recall that 0 ∈ C ).Applying Hahn-Banach Theorem, find a linear functional v ∗ ∈ X ∗ such that v ∗ ( tx ) > sup v ∗ ( C ) ≥
0. Then v ∗ ∈ V ∗ C and v ∗ ( x ) >
0, which contradicts the property of x established at the end of the preceding paragraph. Thiscontradiction shows that x ∈ V C and then x ∗ ( x ) > x ∗ ( V C ) = ∞ , which contradicts the choice ofthe functional x ∗ ∈ V ∗ V C . This contradiction completes the proof of the inclusion V ∗ V C ⊂ cl ∗ ( V ∗ C ). (cid:3) The following lemma implies that polyhedral convex sets in Banach spaces lie on positive Hausdorff distancefrom their characteristic cones.
Lemma 2.6.
For a normed space X , functionals f , . . . , f n : X → R , a vector a = ( a , . . . , a n ) ∈ R n withnon-negative coordinates, and the polyhedral convex set P a = n \ i =1 f − i (cid:0) ( −∞ , a i ] (cid:1) (1) the characteristic cone V P a of the convex set P a coincides with the polyhedral cone P ; (2) d H ( P a , P ) ≤ d H ( P , P ) · max ≤ i ≤ n a i ,where = (0 , . . . , and = (1 , . . . , .Proof. We consider the space R n as a Banach lattice with coordinatewise operations of minimum and maximum.1. The first statement is easy and is left to the reader as an exercise.2. To prove the second statement, we first check that d H ( P , P ) < ∞ . By Lemma 2.1(2), P = conv( F )+cone( E )for some finite sets F, E ⊂ X . It follows that the cone cone( E ) coincides with the characteristic cone P of P andhence P = conv( F ) + P . Then d H ( P , P ) ≤ d H (conv( F ) + P , P ) ≤ d H (conv( F ) , { } ) < ∞ . Let a = max ≤ i ≤ n a i . Taking into account that that the norm of the Banach space X is homogeneous and that P ⊂ P a ⊂ P a , we get the required inequality d H ( P a , P ) ≤ d H ( P a , P ) = a · d H ( P , P ) = d H ( P , P ) · max ≤ i ≤ n a i < ∞ . (cid:3) Recognizing separable components of
Conv H ( X )In this section we shall prove some lemmas that will help us to recognize closed convex sets C with separablecomponent H C . First we consider the finite-dimensional case. The following lemma was proved by V.Klee in [8].However, we give an alternative proof based on a Ramsey-theoretic argument. Lemma 3.1.
If the component H C of a closed convex subset C of a finite-dimensional Banach space X contains apolyhedral convex set, then H C contains a countable dense family of polyhedral convex sets.Proof. The case C = X is trivial because in this case the component H C = { X } contains a unique convex set X ,which is polyhedral as the intersection of the empty family of closed half-spaces. So, we assume that H C containssome polyhedral convex set P = X . Without loss of generality we can assume that 0 ∈ P . By Lemma 2.4, d H ( C, P ) < ∞ implies V C = V P = X . N APPROXIMATIVELY POLYHEDRAL CONVEX SETS 5
Write P as a finite intersection of closed half-spaces P = k \ i =1 f − i (cid:0) ( −∞ , a i ] (cid:1) where f , . . . , f k : X → R are linear continuous functionals with unit norm and a , . . . , a k are some real numbers.It follows from 0 ∈ P that a , . . . , a n ≥
0. According to Lemma 2.6, we lose no generality assuming that a = · · · = a k = 0, which implies that P is a polyhedral cone that coincides with its characteristic cone V P = V C . ByLemma 2.1, P = cone( B ) for some finite subset B ⊂ X .By our assumption, the Banach space X is finite-dimensional and hence separable. So, we can fix a countabledense subset D ⊂ X . Next, for every finite subset F ⊂ D consider its convex hull conv( F ) and the polyhedralconvex set C F = conv( F ) + V C = conv( F ) + cone( B ) . It remains to check that the countable family C = { C F : F is a finite subset of D } is dense in H C .Given a convex set A ∈ H and ε >
0, we shall find a finite subset F ⊂ D with d H ( C F , A ) < ε . Denote by¯ B = { x ∈ X : k x k ≤ } the closed unit ball of the Banach space X . Then for every r > r ¯ B = { r · x : x ∈ ¯ B } coincides with the closed r -ball { x ∈ X : k x k ≤ r } . Claim 3.2.
There is r ∈ R + so large that the convex set A r = ( A ∩ r ¯ B ) + P is not empty and lies on the Hausdorffdistance d H ( A r , A ) ≤ ε from A .Proof. It follows from d H ( A, C ) < ∞ that V A = V C = P , see Lemma 2.4. Then for each r ∈ R + we get the inclusion A r = ( A ∩ r ¯ B ) + P ⊂ A + P = A + V A = A. Assuming that d H ( A r , A ) > ε for all r ∈ R + , we can construct an increasing sequence of positive real numbers( r n ) n ∈ ω and a sequence of points ( x n ) n ∈ ω in A such that k x n k ≤ r n and dist ( x n +1 , A r n ) > ε for all n ∈ ω .Consequently, for every n < m we get( x m + ε ¯ B ) ∩ ( x n + P ) ⊂ ( x m + ε ¯ B ) ∩ ( A r n + P ) = ∅ , which implies x m − x n / ∈ ε ¯ B + P .Let us recall that P = T ki =1 H i where H i = f − i (cid:0) ( −∞ , (cid:1) for i ≤ k . Using Lemma 2.6, we can choose δ > T ki =1 f − i (cid:0) ( −∞ , δ ] (cid:1) ⊂ P + ε B .It follows that for any n < m we get x m − x n / ∈ T ni =1 f − i (cid:0) ( −∞ , δ ] (cid:1) and hence there is a number i = i ( n, m ) ∈{ , . . . , k } such that x m − x n / ∈ f − i (cid:0) ( −∞ , δ ] (cid:1) and hence f i ( x m ) > f i ( x n ) + δ .The correspondence i : ( n, m ) → i ( n, m ) can be thought as a finite coloring of the set [ ω ] = { ( n, m ) ∈ ω : n < m } of pairs of positive integers. The Ramsey Theorem 5 of [5] yields an infinite subset Ω ⊂ ω and a number i ∈ { , . . . , k } such that i ( n, m ) = i and hence f i ( x m ) > f i ( x n ) + δ for all numbers n < m in Ω. This impliessup c ∈ C f i ( c ) ≥ sup n ∈ Ω f i ( x n ) = ∞ , which is not possible as sup f i ( C ) ≤ (sup f i ( P ))+ d H ( C, P ) = d H ( C, P ) < ∞ . (cid:3) Claim 3.2 yields us a real number r ∈ R + such that the intersection A ∩ r ¯ B is not empty and dist ( A r , A ) < ε where A r = ( A ∩ r ¯ B ) + P . By [6], the compact convex set A ∩ r ¯ B can be approximated by a finite subset F ⊂ D such that d H (conv( F ) , A ∩ r ¯ B ) < ε . Then the polyhedral convex set C F = conv( F ) + P lies on Hausdorff distance d H ( C F , A r ) < ε from the set A r and hence on Hausdorff distance d H ( C F , A ) ≤ d H ( C F , A r ) + d H ( A r , A ) < ε from A . (cid:3) In order to generalize Lemma 3.1 to infinite-dimensional Banach spaces X , we now establish some simple prop-erties of maps between spaces of closed convex sets, induced by quotient operators.Let us recall that for a Banach space ( X, k · k X ) and a closed linear subspace Z ⊂ X , the quotient Banach space Y = X/Z is endowed with the norm k y k Y = inf (cid:8) k x k X : x ∈ q − ( y ) (cid:9) , where q : X → Y , q : x x + Z , stands for the quotient operator.The quotient operator q : X → Y induces an operator ¯ q : Conv H ( X ) → Conv H ( Y ) assigning to each closedconvex set C ⊂ X the closure ¯ qC of its image qC in the Banach space Y . The following lemma is simple and is leftto the reader as an exercise. TARAS BANAKH AND IVAN HETMAN
Lemma 3.3.
Let Z be a closed linear subspace of a Banach space X , Y = X/Z be the quotient Banach space, and q : X → Y be the quotient operator. (1) A convex set C ⊂ X with Z ⊂ V C is closed in X if and only if the image qC is closed in Y . (2) A convex set C ⊂ X with Z ⊂ V C is polyhedral in X if and only if its image qC is polyhedral in Y . (3) For any non-empty convex sets
A, B ⊂ X with Z ⊂ V A ∩ V B we get d H ( A, B ) = d H ( qA, qB ) . Now we are able to prove an (infinite-dimensional) generalization of Lemma 3.1, which will be used in the proofof the implications (3) ⇒ (1 ,
5) from Theorem 1.1.
Lemma 3.4.
If the component H C of a non-empty closed convex subset C of a Banach space X contains a polyhedralconvex set, then H C contains a countable dense family of polyhedral closed sets, which implies that the space H C isseparable and the convex set C is approximatively polyhedral.Proof. The statement of the lemma is trivial if C = X . So, we assume that C = X and H C contains a polyhedralconvex set P . Replacing P by its shift, we can assume that 0 ∈ P . Write P as a finite intersection of closedhalf-spaces P = k \ i =1 f − i (cid:0) ( −∞ , a i ] (cid:1) , where f , . . . , f k : X → R are suitable linear continuous functionals and a , . . . , a k are suitable non-negative realnumbers. It follows from d H ( C, P ) < ∞ that the characteristic cone V C = V P = k \ i =1 f − i (cid:0) ( −∞ , (cid:1) is polyhedral and the closed linear subspace Z = − V C ∩ V C = k \ i =1 f − i (0)has finite codimension in X .Then the quotient Banach space Y = X/Z is finite-dimensional. Taking into account that Z ⊂ V P ∩ V C andapplying Lemma 3.3(1,3), we conclude that the images qC and qP are closed convex sets in Y with d H ( qC, qP ) < ∞ .Moreover, the convex set qP is polyhedral in Y . Since the Banach space Y is finite-dimensional, it is legal to applyLemma 3.1 in order to find a dense countable subset D Y ∈ H qC that consists of polyhedral convex sets. ByLemma 3.3(2), the countable family D X = { q − ( D ) : D ∈ D X } consists of polyhedral convex subsets of X and byLemma 3.3(3) is dense in the component H C of C . (cid:3) Recognizing non-separable components of
Conv H ( X )In this section we develop some tools for recognizing non-separable components of the space Conv H ( X ). Lemma 4.1.
Let C be a convex subset of a linear space X and a, b ∈ X two points such that [ a, b ] ∩ C = ∅ . Thenfor any points x ∈ conv( C ∪ { a } ) and y ∈ conv( X ∪ { b } ) we get [ x, y ] ∩ C = ∅ .Proof. The lemma trivially holds if x or y belongs to the set C . So, we assume that x, y / ∈ C . It follows that thepoint x ∈ conv( C ∪ { a } ) \ C can be written as x = t x a + (1 − t x ) c x for some t x ∈ (0 ,
1] and c x ∈ C . The same is truefor the point y ∈ conv( C ∪ { b } ) \ C which can be written as y = t y b + (1 − t y ) c y for some t y ∈ (0 ,
1] and c y ∈ C .By our assumption, the intersection [ a, b ] ∩ C contains some point c = ta + (1 − t ) b where t ∈ [0 , x, y ] meets the convex hull conv( { c, c x , c y } ) ⊂ C of the 3-element set { c, c x , c y } ⊂ C . This will follow as soon as we find real numbers u, α, α x , α y ∈ [0 ,
1] such that α + α x + α y = 1 and αc + α x c x + α y c y = ux + (1 − u ) y = u ( t x a + (1 − t x ) c x ) + (1 − u )( t y b + (1 − t y ) c y ) . The number u ∈ [0 ,
1] can be found from the equation ut x a + (1 − u ) t y b = ( ut x + (1 − u ) t y )( ta + (1 − t ) b ) = ( ut x + (1 − u ) t y ) c, which has a well-defined solution u = t · t y t · t y + (1 − t ) t x . N APPROXIMATIVELY POLYHEDRAL CONVEX SETS 7
The remaining numbers α, α x and α y can be found as α = ut x + (1 − u ) t y , α x = u (1 − t x ) , α y = (1 − u )(1 − t y ) . (cid:3) The following lemma will be used for the proof of the implication (6) ⇒ (4) from Theorem 1.1. Lemma 4.2.
The component H C ⊂ Conv H ( X ) of a closed convex subset C of a Banach space X has density dens( H C ) ≥ c provided H C contains a positively hiding closed convex subset P of X .Proof. Since H C = H P , we lose no generality assuming that the convex set C itself is positively hiding, whichmeans that there is an infinite subset A ⊂ X \ C on positive distance ε = inf a ∈ A dist ( a, C ) from C , which is hiddenbehind the set C in the sense that for any distinct points a, b ∈ A the affine segment [ a, b ] meets the set C .Fix any point c ∈ C and for every point a ∈ A choose a point b a ∈ [ c , a ] on the distance dist ( b a , C ) = ε from C . The choice of the point b a is possible as dist ( a, C ) ≥ ε . Lemma 4.1 guarantees that the set B = { b a : a ∈ A } is infinite and is hidden behind the set C . Moreover, this set lies in the 2 ε -neighborhood C + 2 ε B of C . Here B = { x ∈ X : k x k < } stands for the open unit ball in the Banach space X .Now for any subset β ⊂ B consider the convex set C β = conv( C ∪ β ). Applying Lemma 4.1 one can show thatthis set is closed in X and C β = S b ∈ β conv( C ∪ { b } ). Taking into account that C ⊂ C β ⊂ C + 2 ε B , we see that d H ( C, C β ) ≤ ε and hence C β belongs to the component H C of C .We claim that for any two distinct subsets α, β ⊂ B , the convex sets C α and C β lie on the Hausdorff distance d H ( C α , C β ) ≥ ε . Since α = β , there is a point b ∈ ( β \ α ) ∪ ( α \ β ). We lose no generality assuming that b ∈ β \ α .Then b ∈ C β and dist ( b, C α ) ≥ ε . Indeed, assuming that dist ( b, C α ) < ε , we conclude that the open ε -ball b + ε B meets the set C α = conv( C ∪ α ) = S a ∈ α conv( C ∪ { a } ) at some point x that belongs to conv( C ∪ { a } ) for somepoint a ∈ α . Since the set B ∋ a, b is hidden behind the set C , the segment [ a, b ] meets the set C . By Lemma 4.1,the segment [ x, b ] also meets the set C , which is not possible as [ x, b ] lies in the ε -ball b + ε B , which does not meet C as dist ( b, C ) = ε . This contradiction shows that dist ( b, C α ) ≥ ε and hence d H ( C β , C α ) ≥ ε .Now we see that the component H C contains the subset C = { C β : β ⊂ B } of cardinality |C| ≥ | B | ≥ c ,consisting of points on mutual distance ≥ ε . This implies that dens( H C ) ≥ |C| ≥ c . (cid:3) Recognizing infinitely hiding convex sets
In this section we develop some tools for recognizing infinitely hiding convex sets. In fact, we shall work withthe following relative version of this property.Let C , C be two convex sets in a Banach space X . We shall say that C is C -infinitely hiding if C hides someinfinite set A ⊂ aff( C ) such that sup a ∈ A dist ( a, C ) = ∞ .It is easy to see that a convex set C ⊂ X is infinitely hiding if and only if it is C -infinitely hiding.We start with the following elementary lemma. Lemma 5.1.
Let C ∋ be a convex set in a Banach space and V ¯ C be the characteristic cone of its closure. For alinear subspace Z ⊂ X , the intersection Z ∩ V ¯ C is C -infinitely hiding if the cone Z ∩ V ¯ C is a hiding convex set in Z .Proof. Assume that the cone Z ∩ V ¯ C hides some infinite injectively enumerated set { a n } n ∈ ω ⊂ Z \ V ¯ C . By Lemma 2.3,for every n ∈ ω , there is a real number t n > dist ( t n a n , C ) > n . It is clear that for the set A = { t n a n } n ∈ ω we get sup a ∈ A dist ( a, C ) = lim n →∞ dist ( t n a n , C ) = ∞ . It remains to show that for every distinct numbers n, m ∈ ω thesegment [ t n a n , t m b m ] intersects the cone Z ∩ V ¯ C .Since the set { a n , a m } ⊂ A ⊂ Z is hidden behind Z ∩ V ¯ C , the segment [ a n , a m ] meets the cone Z ∩ V ¯ C at somepoint c = τ a n + (1 − τ ) a m where τ ∈ [0 , u = τ t m τ t m + (1 − τ ) t n ∈ [0 , ut n a n + (1 − u ) t m a m = t n t n τ t m + (1 − τ ) t n ( τ a n + (1 − τ ) a m ) = t n t m τ t m + (1 − τ ) t n c ∈ [ t n a n , t m a m ] ∩ V ¯ C and hence the intersection Z ∩ V ¯ C ∩ [ t n a n , t m a m ] ∋ ut n a n + (1 − u ) t m a m is not empty. (cid:3) By [1], a closed convex subset C of a complete linear metric space X is hiding if and only if C is polyhedral inits closed linear hull. This characterization combined with Lemma 5.1 implies: TARAS BANAKH AND IVAN HETMAN
Lemma 5.2.
Let C ∋ be a convex set in a Banach space and V ¯ C be the characteristic cone of its closure. For aclosed linear subspace Z ⊂ X the intersection V = Z ∩ V ¯ C is infinitely C -hiding if the cone V is polyhedral in itsclosed linear hull V ± = cl( V − V ) . This lemma implies its absolute version:
Lemma 5.3.
A closed convex subset C of a Banach space is infinitely hiding if its characteristic cone V C is notpolyhedral in its closed linear hull V ± C = cl( V C − V C ) . Next, we derive the infinite hiding property of a convex set from that property of its projections. We start withthe following algebraic fact.
Lemma 5.4.
Let q : X → ˜ X be a linear operator between linear spaces, E = q − (0) be the kernel of q , and C ⊂ X be a convex set such that V C ∩ E − V C ∩ E = E . If the image ˜ C = q ( C ) hides some countable set ˜ A ⊂ aff( ˜ C ) , then C hides some set A ⊂ aff( C ) with q ( A ) = ˜ A .Proof. Let ˜ A = { ˜ a n : n ∈ ω } be an injective enumeration of the countable set ˜ A . By induction, for every n ∈ ω weshall choose a point a n ∈ q − (˜ a n ) ∩ aff( C ) so that [ a n , a m ] ∩ C = ∅ for every numbers n < m , and a n ∈ C if ˜ a n ∈ ˜ C .We start the inductive construction choosing any point a ∈ q − (˜ a ) ∩ aff( C ). Such point a exists since q (aff( C )) = aff( ˜ C ). If ˜ a ∈ ˜ C , then we can additionally assume that a ∈ C . Assume that for some n ≥ a , . . . , a n − have been constructed. We need to choose a point a n ∈ q − (˜ a n ) ∩ aff( C ) so that [ a i , a n ] ∩ C = ∅ for all i < n . If ˜ a n ∈ ˜ C , then let a n ∈ C be any point with q ( a n ) = ˜ a n . So, assume that ˜ a n / ∈ ˜ C . Let I n = { i ∈ ω : i < n, ˜ a i / ∈ ˜ C } .Since the set ˜ A ⊂ ˜ X is hidden behind the convex set ˜ C , for every i ∈ I n the intersection [˜ a i , ˜ a n ] ∩ ˜ C containssome point ˜ c i which can be written as the convex combination ˜ c i = u i ˜ a i + (1 − u i )˜ a n for some u i ∈ (0 , c i ∈ ˜ C , there is a point c i ∈ C with q ( c i ) = ˜ c i . It follows from ˜ a n = ˜ c i − u i ˜ a i − u i that the point a ′ i = c i − u i a i − u i belongs tothe preimage q − (˜ a n ).It follows from E = V E ∩ C − V E ∩ C that the intersection T i ∈ I n ( a ′ i + V E ∩ C ) contains some point a n . For this pointwe get u i a i + (1 − u i ) a n ∈ c i + V C ⊂ C and hence [ a i , a n ] ∩ C = ∅ for all i < n , which completes the inductive step.After completing the inductive construction, we obtain the countable set A = { a n } n ∈ ω that has the requiredproperty. (cid:3) Lemma 5.4 implies its C -infinitely hiding version. Lemma 5.5.
Let X be a Banach space, E be a closed linear subspace of X , ˜ X = X/E be the quotient Banachspace and q : X → ˜ X be the quotient operator. Let C , C be two convex sets in X and ˜ C = q ( C ) , ˜ C = q ( C ) betheir quotient images in ˜ X . The convex set C is C -infinitely hiding X if its image ˜ C is ˜ C -infinitely hiding and E = V E ∩ C − V E ∩ C .Proof. If ˜ C is ˜ C -infinitely hiding, then ˜ C hides some infinite set ˜ A ⊂ aff( ˜ C ) such that sup ˜ a ∈ ˜ A dist (˜ a, ˜ C ) = ∞ .By Lemma 5.4, there is a set A ⊂ aff( C ) with q ( A ) = ˜ A , hidden behind the set C .Since the quotient operator q : X → ˜ X has norm k q k ≤
1, for every point a ∈ A and its image ˜ a = q ( a ), we get dist (˜ a, ˜ C ) ≤ dist ( a, C ). Consequently, sup a ∈ A dist ( a, C ) ≥ sup ˜ a ∈ ˜ A dist (˜ a, ˜ C ) = ∞ , which means that the set C is C -infinitely hiding. (cid:3) The preceding lemma allows us to derive the C -infinite hiding property of a convex set from that property of itsprojection. Our next lemma will help us to do the same using the C -infinite hiding property of two-dimensionalsections of the convex set. Lemma 5.6.
Let C be a closed convex subset of a Banach space X and Z be a two-dimensional linear subspace of X such that the convex set C ∩ Z has non-empty interior C in Z , which contains zero. If d H ( C , V C ) = ∞ , thenthe convex set C is C -infinitely hiding.Proof. The equality d H ( C , V C ) = ∞ implies that the open convex subset C of the plane Z is not bounded.Consequently, its characteristic cone V C = V ¯ C = Z ∩ V C is unbounded too. Moreover, the cone V C is not a plane,not a half-plane, and not a line (otherwise C would be on finite Hausdorff distance from its characteristic cone V C ). Consequently, we can choose two linearly independent vectors e , e ∈ Z such that the cone V C is equal tocone( { e } ) or to cone( { e , e } ). Let e ∗ , e ∗ ∈ Z ∗ be the coordinate functionals corresponding to the base e , e of Z .This means that z = e ∗ ( z ) e + e ∗ ( z ) e for each vector z ∈ Z . N APPROXIMATIVELY POLYHEDRAL CONVEX SETS 9 If V C = cone( { e , e } ), then the equality d H ( C , V C ) = ∞ implies that inf e ∗ ( C ) = −∞ or inf e ∗ ( C ) = −∞ .We lose no generality assuming that inf e ∗ ( C ) = −∞ .If V C = cone( { e } ), then the equality d H ( C , V C ) = ∞ implies that inf e ∗ ( C ) = −∞ or sup e ∗ ( C ) = + ∞ .Changing e to − e , if necessary, we can assume that inf e ∗ ( C ) = −∞ .So, in both cases we can assume that inf e ∗ ( C ) = −∞ .By induction, we shall construct a sequence of points ( a n ) n ∈ ω in cone( { e , − e } ) such that for every n ∈ ω thefollowing conditions are satisfied:(1) dist ( a n , C ) > n ;(2) e ∗ ( a n ) > e ∗ ( a n − ) > e ∗ ( a n ) < e ∗ ( a n − ) <
0, and | e ∗ ( a n ) | e ∗ ( a n ) < | e ∗ ( a n − ) | e ∗ ( a n − ) ;(3) [ a n , a k ] ∩ C = ∅ for all k < n .We start the inductive construction selecting a point a ∈ R + · ( e − e ) on the distance dist ( a , C ) > C . Such point a exists because e − e / ∈ V C = Z ∩ V C . Now assume that for some n ∈ N we haveconstructed points a , . . . , a n ∈ cone( { e , − e } ) satisfying the conditions (1)–(3). It follows from inf e ∗ ( C ) = −∞ and e ∈ V C \ ( − V C ) that there exists a point c ∈ C such that e ∗ ( c ) > c ∗ ( a n ) , e ∗ ( c ) < e ∗ ( a n ) and | e ∗ ( c ) | e ∗ ( c ) < | e ∗ ( a n ) | e ∗ ( a n ) . Now consider the vector v = c − a n and observe that v / ∈ V C = V ¯ C . Consequently, c + R + v ¯ C , which allows usto find a point a n +1 ∈ c + R + v with dist ( a n +1 , C ) > n + 1 (using Lemma 2.3). It can be shown that the point a n +1 satisfies the condition (2). Since the segment [ a n , a n +1 ] contains the point c , it meets the set C ∩ Z .It remains to check that [ a k , a n +1 ] ∩ C = ∅ for every number k < n . By the inductive assumption, the segment[ a k , a n ] meets the set C at some point c ′ . r ✲ e ✻ e (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) V Z ∩ C ◗◗◗◗◗◗◗◗ r c ′ ❛❛❛❛❛❛❛❛❛❛❛ r a k r a n ❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛ r c ❳❳❳❳❳❳❳❳❳❳❳❳❳❳ ❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳ ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ r a n +1 By an elementary plane geometry argument one can prove that the segment [ a k , a n +1 ] meets the triangleconv( { , c ′ , c } ) ⊂ C and hence meets the set C . This completes the inductive step.After completing the inductive construction, we obtain the infinite set A = { a n } n ∈ ω with sup a ∈ A dist ( a, C ) = ∞ which is hidden behind C . This means that the set C is C -infinitely hiding. (cid:3) Lemma 5.7.
Let C be a convex subset of a Banach space X and Z be a finite-dimensional linear subspace of X such that the convex set C ∩ Z has non-empty interior C in Z and ∈ C . If d H ( C , V C ) = ∞ , then the convexset C is C -infinitely hiding.Proof. This lemma will be proved by induction on the dimension dim( Z ) of Z . The lemma is trivially true ifdim( Z ) ≤
1. Assume that the lemma has been proved for all triples (
X, C, Z ) with dim( Z ) < n . Now we prove thislemma for dim( Z ) = n . Assuming that d H ( C , V C ) = ∞ , we need to prove that the convex set C is C -infinitelyhiding. To derive a contradiction, assume that C is not C -infinitely hiding. In this case Lemma 5.6 implies thefollowing fact, which will be used several times in the subsequent proof. Claim 5.8.
For each two-dimensional linear subspace Z ⊂ Z we get d H ( Z ∩ C , V Z ∩ C ) < ∞ . Now consider the characteristic cone V C = V ¯ C of the open convex set C in Z . Claim 5.9.
The linear subspace − V C ∩ V C is trivial.Proof. Assume that the linear subspace E = − V C ∩ V C is not equal to { } . Consider the quotient Banach space˜ X = X/E , the quotient operator q : X → ˜ X , the convex set ˜ C = q ( C ), and the finite-dimensional subspace˜ Z = q ( Z ) of dimension dim( ˜ Z ) < dim( Z ) = n . Since the quotient operator q | Z : Z → ˜ Z is open, the convex set˜ C = q ( C ) is open in ˜ Z and hence the convex set ˜ Z ∩ ˜ C has non-empty interior in ˜ Z . Since E ⊂ V C , the set ˜ C isclosed in the Banach space ˜ X by Lemma 3.3(1). Now we can see that the triple ( ˜ X, ˜ C, ˜ Z ) satisfies the requirementsof Lemma 5.7 with dim( ˜ Z ) < dim( Z ) = n . So, by the inductive assumption, the convex set ˜ C is ˜ C -infinitelyhiding.Since E = V E ∩ C = V E ∩ C − V E ∩ C , we can apply Lemma 5.4 to conclude that the open convex set C is C -infinitely hiding in X . But this contradicts our assumption. (cid:3) Claim 5.10. dim( V C ) ≥ .Proof. Assume conversely that dim( V C ) ≤
1. The equality d H ( C , V C ) = ∞ implies that the open convex subset C is unbounded in the finite-dimensional linear space Z ∩ C and consequently V C = 0. Since − V C ∩ V C = { } ,we conclude that V C = ¯ R + e for some non-zero vector e ∈ V C . Now consider the linear subspace E = R e ⊂ Z , thequotient Banach space ˜ X = X/E , the finite-dimensional linear subspace ˜ Z = q ( Z ), the convex sets ˜ C = q ( C ), andthe open convex set ˜ C = q ( C ), which is dense in ˜ C . Claim 5.8 guarantees that the set ˜ C has trivial characteristiccone and hence is bounded in the finite-dimensional space ˜ Z . This implies that d H ( C , V C ) < ∞ , which is a desiredcontradiction. (cid:3) Since C is not C -infinitely hiding, Lemma 5.2, guarantees that the characteristic cone V C is polyhedral in Z and hence it can be written as a finite intersection of closed half-spaces V C = k \ i =1 f − i (cid:0) ( −∞ , (cid:1) determined by some linear functionals f , . . . , f k : Z → R . We shall assume that the number k in this representationis the smallest possible.It follows from d H ( C , V C ) = ∞ and Lemma 2.6 that sup f i ( C ) = ∞ for some i ≤ k . Claim 5.11.
The face f − i (0) ∩ V C of V C contains a non-zero vector e .Proof. By the minimality of k the cone V = T { f − j (cid:0) ( −∞ , (cid:1) : 1 ≤ j ≤ k, j = i } is strictly larger than V C andhence contains a point x ∈ V \ V C . For this point x we get f j ( x ) ≤ j = i , and f i ( x ) > V C ) ≥
2, there exists a vector y ∈ V C \ R x . Such choice of x guarantees that 0 / ∈ [ x, y ]. Since f i ( y ) ≤ f i ( x ) >
0, there is a point e ∈ [ x, y ] with f i ( e ) = 0. For every j = i , the inequalities f j ( x ) ≤ f j ( y ) ≤ f j ( e ) ≤
0. Conseqeuntly, e is a required non-zero vector in f − i (0) ∩ V C . (cid:3) Claim 5.11 yields a non-zero vector e ∈ f − i (0) ∩ V C . Consider the 1-dimensional linear subspace E = R e of X and let ˜ X = X/E be the quotient Banach space. Observe that ˜ X contains the finite-dimensional linear subspace˜ Z = Z/E of dimension dim( ˜ Z ) = dim( Z ) − < n . Let q : X → ˜ X be the quotient operator, and ˜ C = q ( C ),˜ C = q ( C ) be the images of the convex sets C and C in ˜ X . It follows from E = q − (0) ⊂ Z that ˜ Z ∩ ˜ C = q ( Z ∩ C )and ˜ C = q ( C ) coincides with the interior of the set q ( Z ∩ C ) = ˜ Z ∩ ˜ C in ˜ Z . So, the triple ( ˜ X, ˜ Z, ˜ C ) satisfies theassumptions of the lemma.We claim that d H ( ˜ C , V ˜ C ) = ∞ . Since E ⊂ f − i (0), there is a linear functional ˜ f i : ˜ Z → R such that f i = ˜ f i ◦ q | Z . Claim 5.12. V ˜ C ⊂ ˜ f − i ( −∞ , .Proof. Assume conversely that the characteristic cone V ˜ C contains some vector w ∈ ˜ Z with ˜ f i ( w ) >
0. Then R + w ⊂ ˜ Z . Pick any vector v ∈ q − ( w ) ⊂ Z and consider the two-dimensional subspace Z = lin( { v, e } ) spanning thevectors v and e . Observe that for every t ∈ R we get f i ( v + te ) = ˜ f i ( w ) >
0, which implies that ( v + t R ) ∩ V Z ∩ C = ∅ .Then V Z ∩ C ⊂ R e − ¯ R + v and q ( V Z ∩ C ) ⊂ − ¯ R + w . On the other hand, the projection q ( Z ∩ C ) contains the half-line R + w , which implies that d H ( Z ∩ C, V Z ∩ C ) = ∞ . But this contradicts Claim 5.8. (cid:3) Taking into account that ∞ = sup f i ( C ) = sup ˜ f i ( ˜ C ), we conclude that d H ( ˜ C , V ˜ C ) = ∞ and by the inductiveassumption, the open convex set ˜ C is ˜ C -infinitely hiding (as dim( ˜ Z ) < dim( Z ) = n ). Since E = R + e − R + e = V E ∩ C − V E ∩ C , we can apply Lemma 5.4 to conclude that the open convex set C is C -infinitely hiding in X . Thiscontradiction completes the proof of Lemma 5.7. (cid:3) N APPROXIMATIVELY POLYHEDRAL CONVEX SETS 11
Lemma 5.7 admits the final (and main) lemma of this section.
Lemma 5.13.
A closed convex subset C of a Banach space X is infinitely hiding if d H ( A ∩ C, A ∩ V C ) = ∞ forsome finite-dimensional affine subspace A ⊂ X .Proof. It is well-known that C ∩ A has non-empty interior C in its affine hull aff( C ∩ A ). Moreover, C ∩ A coincideswith the closure ¯ C of C in A . Shifting the set C , if necessary, we can assume that 0 ∈ C . Then Z = aff( C ) isa finite-dimensional linear subspace of X such that C ∩ Z = C ∩ A has non-empty interior C which contains zeroand is dense in C ∩ Z . It follows that d H ( C , V C ) = d H ( Z ∩ C, V Z ∩ C ) = ∞ and hence C is C -infinitely hiding byLemma 5.7, and C is infinitely hiding as C ⊂ C . (cid:3) Approximating by positively hiding convex sets
In this section we search for conditions guaranteeing that a closed convex subset C of a Banach space can beapproximated by positively hiding convex subsets of X .At first we construct biorthogonal sequences, which are related to convex sets that have trivial characteristiccone.We recall that a sequence of pairs { ( x n , x ∗ n ) } n ∈ ω ⊂ X × X ∗ is biorthogonal if x ∗ n ( x n ) = 1 and x ∗ n ( x k ) = 0 for all n = k , see [7, 1.1]. Lemma 6.1.
Assume that a closed convex subset C of an infinite-dimensional Banach space X has trivial charac-teristic cone V C = { } . Then there exists a biorthogonal sequence { ( x n , x ∗ n ) } n ∈ ω ⊂ X × V ∗ C such that k x ∗ n k = 1 ≤k x n k < for all n ∈ ω .Proof. Replacing C by a shift of its closed neighborhood, we can assume that the convex subset C of X hasnon-empty interior C , which contains zero. After such replacement the cones V C and V ∗ C will not change, seeLemma 2.4.The biorthogonal sequence { ( x n , x ∗ n ) } n ∈ ω will be constructed by induction. We start by choosing an arbitraryfunctional x ∗ ∈ V ∗ C and a point x ∈ X with 1 = k x ∗ k = x ∗ ( x ) ≤ k x k < k ∈ ω a finite biorthogonal sequence { ( x n , x ∗ n ) } n
4. Assuming the converse, find a functional x ∗ ∈ X ∗ with unit norm such that x ∗ ( x k ) = k x k k ≥
4. Then the functional y ∗ = x ∗ k − x ∗ belongs to the ball B ∗ and thus y ∗ ( x k ) >
0. On the otherhand, y ∗ ( x k ) = x ∗ k ( x k ) − x ∗ ( x k ) ≤ − · (cid:3) Lemma 6.2.
Assume that a closed convex subset C of an infinite-dimensional Banach space X has trivial charac-teristic cone V C = { } . Then for each ε > there is a positively hiding closed convex set C ε ⊂ X with d H ( C ε , C ) ≤ ε .Proof. By Lemma 6.1, there exists a biorthogonal sequence { ( x n , x ∗ n ) } n ∈ ω ⊂ X × V ∗ C such that 1 = k x ∗ n k ≤ k x n k < n ∈ ω .Then for every ε >
0, a positively hiding convex set C ε with d H ( C ε , C ) ≤ ε can be defined by the formula: C ε = (cid:8) x ∈ cl( C + ε B ) : ∀ n ∈ ω x ∗ n ( x ) ≤ ε + sup x ∗ n ( C ) (cid:9) where B = { x ∈ X : k x k < } stands for the open unit ball of the Banach space X . It is clear that C ⊂ C ε ⊂ cl( C + ε B ), which implies that d H ( ˜ C, C ) ≤ ε . It remains to check that the set ˜ C is positively hiding. This can bedone as follows.For every n ∈ ω choose a point c n ∈ C with x ∗ n ( c n ) > sup x ∗ n ( C ) − ε and consider the point a n = c n + ε x n .We claim that dist ( a n , C ε ) ≥ ε . Indeed, for any point c ∈ C ε , we get x ∗ n ( c ) ≤ sup x ∗ n ( C ) + ε while x ∗ n ( a n ) = ε x ∗ n ( x n ) + x ∗ n ( c n ) > ε x ∗ n ( C ) − ε
16 = 3 ε
16 + sup x ∗ n ( C ) . Consequently, k a ∗ n − c k = k x ∗ n k · k a n − c k ≥ x ∗ n ( a n ) − x ∗ ( c ) ≥ ε − ε = ε .So, the set A = { a n } n ∈ ω lies on positive distance inf a ∈ A dist ( a, C ε ) ≥ ε from C ε . To show that the set A isinfinite and hidden behind C ε , it suffices to check that for any distinct numbers n, m the midpoint a n + a m ofthe segment [ a n , a m ] belongs to the convex set C ε . Taking into account that a n , a m ⊂ cl( C + ε B ), we conclude that a n + a m ∈ [ a n , a m ] ⊂ cl( C + ε B ). The inclusion a n + a m ∈ C ε will follow from definition of C ε as soon as wecheck that x ∗ k ( a n + a m ) ≤ sup x ∗ k ( C ) + ε for every k ∈ ω .If k / ∈ { n, m } , then x ∗ k ( x n ) = x ∗ k ( x m ) = 0 and hence x ∗ k ( a n + a m ) = x ∗ k ( c n + c m ) ≤ sup x ∗ k ( C ) . If k = n , then x ∗ k ( a n + a m ) = x ∗ n ( c n + c m ) + εx ∗ n ( x n ) ≤ sup x ∗ n ( C ) + ε. By analogy we can treat the case k = m . (cid:3) Lemma 6.3.
Assume that for a closed convex subset C of a Banach space X the closed linear subspace Z =cl( V C − V C ) has infinite codimension in X . Then for each ε > there is a positively hiding convex set C ε ⊂ X with d H ( C ε , C ) ≤ ε .Proof. Using Lemma 3.3 (by analogy with the proof of Lemma 3.4), we can reduce the proof to the case − V C ∩ V C = { } . So, from now on we assume that − V C ∩ V C = { } . Replacing C by a shift of its closed neighborhood, wecan assume that C has non-empty interior C in X and 0 ∈ C . If the characteristic cone V C is not polyhedral inits closed linear hull Z = cl( V C − V C ), then by Lemma 5.3, the convex set C is infinitely (and hence positively)hiding and hence we can put C ε = C . So, we assume that the characteristic cone V C is polyhedral in Z . Since − V C ∩ V C = { } , the polyhedrality of V C implies that the closed linear space Z = cl( V C − V C ) is finite-dimensionaland coincides with V C − V C . Now consider the quotient Banach space ˜ X = X/Z , the quotient operator q : X → ˜ X ,and the convex set ˜ C = q ( C ). Since the operator q is open, the image ˜ C = q ( C ) of the interior C of C coincideswith the interior of ˜ C . Now consider the characteristic cone V ˜ C of the open convex set ˜ C .If V ˜ C contains some non-zero vector ˜ v , then for any vector v ∈ q − ( v ) and for the finite-dimensional linearsubspace E = lin( Z ∪ { v } ) the intersection E ∩ C lies on infinite Hausdorff distance d H ( E ∩ C , V E ∩ C ) = ∞ fromits characteristic cone. This is so because q ( V E ∩ C ) ⊂ q ( V C ) = { } while q ( E ∩ C ) ⊃ ¯ R + ˜ v . Then Lemma 5.13guarantees that the set C is infinitely (and hence positively) hiding. In this case we can put C ε = C .So, it remains to consider the case V ˜ C = { } . In this case, Lemma 6.2 yields a positively hiding closed convex set˜ C ε ⊂ ˜ X with d H ( ˜ C ε , ˜ C ) < ε . Now consider the convex set C ε = ( C + ε B ) ∩ q − ( ˜ C ε ) and observe that d H ( C ε , C ) ≤ ε and q ( C ε ) = ˜ C ε .The set ˜ C ε , being positively hiding, hides a countably infinite set ˜ A ⊂ ˜ X \ ˜ C ε on positive distance inf a ∈ ˜ A dist ( a, ˜ C ε ) > C ε . Taking into account that Z = V Z ∩ C − V Z ∩ C and q ( C ε ) = ˜ C ε , with help of Lemma 5.4, we can find an N APPROXIMATIVELY POLYHEDRAL CONVEX SETS 13 infinite subset A ⊂ q − ( ˜ A ) hidden behind the convex set C ε ⊂ C . Since the quotient operator q is not expanding,the set A lies on positive distanceinf a ∈ A dist ( a, C ) = inf a ∈ A dist ( a, C ) ≥ inf ˜ a ∈ ˜ A dist (˜ a, ˜ C ) > C ε . So, C ε is positively hiding. (cid:3) Our next approximation lemma will be used in the proof of the implication (4) ⇒ (2) of Theorem 1.1. Lemma 6.4.
Let C be a closed convex set in a Banach space X . If d H ( C, V C ) = ∞ , then for each ε > there is apositively hiding convex set ˜ C ⊂ X with d H ( ˜ C, C ) ≤ ε .Proof. If the closed linear subspace V ± C = cl( V C − V C ) has infinite codimension in X , then the existence of apositively hiding convex set C ε ⊂ X with d H ( C ε , C ) < ε follows from Lemma 6.3. So, we assume that V ± C has finitecodimension in X . If the characteristic cone V C is not polyhedral in V ± C , then the set C is infinitely (and positively)hiding by Lemma 5.3. In this case we can put C ε = C . It remains to consider the case of polyhedral cone V C in V ± C . Since V ± C has finite codimension in X , the characteristic cone V C is polyhedral in X and hence the closedlinear subspace V ∓ C = − V C ∩ V C has finite codimension in X .Then the quotient Banach space Y = X/V ∓ C is finite-dimensional. Let q : X → Y be the quotient opera-tor. Lemma 3.3 guarantees that d H ( qC, V qC ) = d H ( C, V C ) = ∞ and then the set qC is infinitely hiding in Y byLemma 5.13. By Lemma 5.5, the set C is infinitely (and hence positively) hiding in X . Letting C ε = C , we finishthe proof of the lemma. (cid:3) Proof of Theorem 1.1
To prove the first part of Theorem 1.1, for every closed convex subset C of a Banach space X we should provethe equivalence of the following conditions:(1) C is approximatively polyhedral;(2) the characteristic cone V C of C is polyhedral in X and d H ( C, V C ) < ∞ ;(3) H C contains a polyhedral closed convex set;(4) H C contains no positively hiding closed convex set;(5) the space H C is separable;(6) the space H C has density dens( H C ) < c .It suffices to prove the implications (1) ⇒ (2) ⇒ (3) ⇒ (5) ⇒ (6) ⇒ (4) ⇒ (2) ⇒ (3) ⇒ (1), among which(2) ⇒ (3) and (5) ⇒ (6) are trivial. The remaining implications can be established as follows.To prove the implication (1) ⇒ (2), assume that the closed convex set C is approximatively polyhedral and finda polyhedral convex set P with d H ( C, P ) < ∞ . Lemma 2.4 implies that V C = V P . The polyhedral convex set P canbe written as a finite intersection P = T ni =1 f − i (cid:0) ( −∞ , a i ] (cid:1) of closed hyperplanes determined by some functionals f , . . . , f n : X → R and some real numbers a , . . . , a n ∈ R . It is easy to check that V C = V P = n \ i =1 f − i (cid:0) ( −∞ , (cid:1) , which means that the characteristic cone V C of C is polyhedral. By Lemma 2.6, d H ( P, V C ) = d H ( P, V P ) < ∞ .Consequently, d H ( C, V C ) ≤ d H ( C, P ) + d H ( P, V C ) < ∞ .The implications (3) ⇒ (5) and (3) ⇒ (1) are proved in Lemma 3.4 and (6) ⇒ (4) in Lemmas 4.2. Theimplication (4) ⇒ (2) follows from Lemmas 5.3, 6.3 and 6.4.Next, assuming the Banach space is finite-dimensional we will check that the conditions (1)–(6) are equivalentto: (7) C is not positively hiding;(8) C is not infinitely hiding.It suffices to check that (4) ⇒ (7) ⇒ (8) ⇒ (2). In fact, the first two implications (4) ⇒ (7) ⇒ (8) are trivial,while (8) ⇒ (2) follows from Lemmas 5.3 and 5.13.8. Acknowledgements
The authors would like to express their sincere thanks to Ostap Chervak and Sasha Ravsky for valuable andfruitful discussions, which resulted in finding a correct proof of Lemma 5.13.
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T.Banakh: Ivan Franko National University of Lviv (Ukraine), and Universytet Jana Kochanowskiego, Kielce (Poland)
E-mail address : [email protected] I.Hetman: Department of Mathematics, Ivan Franko National University of Lviv, Universytetska 1, 79000, Ukraine
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