COUNTABLY CONVEX G_{δ} SETS

VLADIMIR FONF AND MENACHEM KOJMAN

Abstract. We investigate countably convex G_{δ} subsets of Banach spaces. A subset of a linear space is countably convex if it can be represented as a countable union of convex sets. A known sufficient condition for countable convexity of an arbitrary subset of a separable normed space is that it does not contain a semi-clique [10]. A semi-clique in a set S is a subset P ⊆ S so that for every x ∈ P and open neighborhood u of x there exists a finite set X ⊆ P ∩ u such that conv(X) ⊈ S. For closed sets this condition is also necessary.

We show that for countably convex G_{δ} subsets of infinite-dimensional Banach spaces there are no necessary limitations on cliques and semi-cliques.

Various necessary conditions on cliques and semi-cliques are obtained for countably convex G_{δ} subsets of finite-dimensional spaces. The results distinguish dimension d ≤ 3 from dimension d ≥ 4: in a countably convex G_{δ} subset of ℝ^{3} all cliques are scattered, whereas in ℝ^{4} a countably convex G_{δ} set may contain a dense in itself clique.

1. Introduction

Let S be a subset of a linear space. Let γ(S) denote the least cardinality of a collection of convex sets, whose union is equal to S. A subset of a linear space is called countably convex if γ(S) ≤ ℵ_{0}, namely, when S = ⋃_{n} C_{n}, where each C_{n} is convex. We are interested in structural properties of countably convex sets, especially in characterization and classification of countably convex subsets of Banach spaces.

Historically, the characterization of unions of convex sets began with Valentine’s paper [14] in which a closed planar with no visually independent subset of size 3 was shown to be the union of 3 convex sets; a subset P ⊆ S is visually independent in S if no two points of P “see” each other through S. Much of the work that followed Valentine dealt with connections between γ and visually independent subsets in closed planar sets ([5, 2, 1, 3, 13, 12]).

There is a natural framework for discussing convexity numbers, which we now present. To every set S in an arbitrary linear space, one associates a hypergraph G(S), called the convexity hypergraph. The vertices of G are the points of S and the hyperedges of G are all finite subsets X ⊆ S with the property conv(X) ⊈ S. A hyperedge of G(S) is called a finite defected subset of S. Much of the geometry of S is lost when passing from S to G(S) (for instance, one cannot tell from G(S) whether three points of S are co-linear or not), but G(S) determines γ(S) because γ(S) is equal to the chromatic number of G(S) [10]. Recall that the chromatic number of a hypergraph is the minimal size of a partition of the set of vertices into free sets, and a set of vertices is free if it does not contain hyperedges. Recall also that a set of vertices is a k-clique if every k-element subset of it is a hyperedge, and is a clique if it is a k-clique for some k ≥ 2.

Countable convexity of closed sets in Banach spaces is characterized by topological properties of G(S):

Definition 1 ([10]). Let S be a subset of a topological linear space. A set P ⊆ S is a semi-clique in S if for every p ∈ P and open neighborhood u ∋ p, there is a finite set X ⊆ P ∩ u so that conv(X) ⊈ S.

A clique is a semi-clique if and only if it is dense in itself, and a semi-clique does not have to be a clique or to contain one. It is readily verified that in every set S the union of all semi-cliques in S is a maximal and closed semi-clique in S, which is called the convexity radical of S, and is denoted K(S).

In [10] it was shown that for every subset S of a separable normed linear space, the set S \K(S) is countably convex. Thus, a subset of a Banach space is countably convex if and only if its radical is coverable by countably many convex subsets of the set. In a closed set S, a non-empty K(S) cannot be covered by countably many convex subsets of S because of the Baire category theorem. Therefore:

Theorem 2 ([10]). A closed subset of a Banach space is countably convex if and only if it does not contain a semi-clique.

In the present paper, semi-cliques and cliques are examined in countably convex G_{δ} subsets of Banach spaces. Theorem 2 above fails badly for G_{δ} subsets of Banach spaces: in every infinite-dimensional Banach space there exists a G_{δ} set S = ⋃ C_{n}, where each C_{n} is convex and dense in S and so that there exists a dense in itself 2-clique P ⊆ S which is dense in S (see Corollary 12 below). This means that although S is countably convex, all of S is contained in K(S), and worse: the radical contains a dense and dense in itself 2-clique.

In a finite-dimensional space, however, the situation is better. The radical of a countably convex G_{δ} set cannot be dense in the set. In fact, it must be nowheredense in the set (Theorem 3 below).

Can there be a dense in itself clique inside the radical of a countably convex G_{δ} subset of a finite-dimensional space? Yes, if d ≥ 4 (Example 4 below); No, if d ≤ 3 (Theorem 5 below).

Finally, can any limitation on the dimension make Theorem 2 hold for G_{δ} sets? The reader can easily check that Theorem 2 holds for G_{δ} subsets of ℝ^{1}, but by Example 6 below it does not hold in any higher dimension.

A word about the method. A G_{δ} subsets of a Banach space satisfies the Baire category theorem; therefore, if it is countably convex and contains a radical [a dense in itself clique], one of its convex components is somewhere dense in the radical [in the dense in itself clique]. The interior of that convex component cannot contain any point from the radical, so the “bad” pattern is contained in the topological boundary of that convex component. This observation reduces the problem to the geometry of cliques which are contained between convex sets and their closure in some G_{δ} set. In a finite dimensional space, a convex set always has a relative interior. This is the source of distinction between finite and infinite dimension.

The precise geometric property which differentiates the behavior of countably convex G_{δ} sets in ℝ^{3} from that in ℝ^{4} comes from the combinatorics of convex polytopes. For every simplicial convex polytope T ⊆ ℝ^{3} (a polytope whose faces are triangles) with ≥ 4k vertices there exists a subset of k vertices any two of which are connected by inner diagonals. However, in ℝ^{4} there exist simplicial polytopes with any prescribed number of vertices and with no inner diagonals at all (see [6], p.61 or Example 4 below).

The results are presented in two sections, which can be read independently of each other. Section 2 deals with G_{δ} sets in finite dimensional spaces and Section 3 is devoted to a construction of “bad” countably convex G_{δ} subsets in every infinitedimensional Banach space.

Our notation is standard. In every normed linear space X, B_{X} and S_{X} denote, respectively, the unit ball and the unit sphere.

2. Finite dimensional spaces

In this Section we investigate countably convex G_{δ} subsets of finite-dimensional spaces. We prove several necessary conditions on the radicals of such sets. In particular, Theorem 5 and Example 4 reveal a surprising dependence of radicals on the dimension of the space.

Theorem 3. If a G_{δ} set S ⊆ ℝ^{n} is countably convex, then the convexity radical of S is nowhere dense in S.

Proof. Assume to the contrary that the theorem fails. Fix a countably convex S ⊆ ℝ^{n} with radical K = K(S) so that there exists an open ball u such that K ∩ u is dense in S ∩ u. Since K is closed in S, K ∩ u = S ∩ u.

Since S = ⋃ C_{k} where each C_{k} is convex, for some k and an open ball v ⊆ u, C_{k} ∩ v is dense in S ∩ v.

Denote C = C_{k} ∩ v. C is convex and is dense in S ∩ v = K ∩ v.

Let H be the affine span of C. Since K has no isolated points, C contains at least two points, and therefore dim H > 0. It holds that S ∩ v = K ∩ v ⊆ clC ⊆ H (the assumption of finite dimensionality is used here, since H is closed in ℝ^{d}). Choose an open ball w so that ∅ ≠ w ∩ C ⊆ int_{H} C. Since S ∩ w ⊆ H and H ∩ w ⊆ C, it holds that K ∩ w ⊆ C, which is a contradiction, since C ⊆ S is convex and K ∩ w contains defected subsets.

The next thing to examine is whether the radical of a finite dimensional countably convex G_{δ} set may contain a dense in itself clique.

Example 4. There is a countably convex G_{δ} set in ℝ^{4} which contains a dense in itself 2-clique.

Proof. In the construction we use the moments curve in ℝ^{4} and its supporting hyperplanes as presented in the construction of cyclic polytope [6], p. 61. For a real number t, let L(t) = (t, t^{2}, t^{3}, t^{4}) ∈ ℝ^{4}, L = {L(t) : t ∈ [0, 1]} and S_{1} = conv(L). Clearly, L is compact, and therefore S_{1} is a compact convex subset of ℝ^{4}.

For any two reals t_{1}, t_{2}, the polynomial (t − t_{1})^{2}(t − t_{2})^{2} = ∑_{0}^{4} a_{i}t^{i} takes strictly positive values at t ≠ t_{1}, t_{2} and 0 at t = t_{1}, t = t_{2}. Define an affine functional φ_{t_{1},t_{2}}(x) = a_{0} + ∑_{1}^{4} a_{i}x_{i}, x = (x_{1}, x_{2}, x_{3}, x_{4}) ∈ ℝ^{4}.

It is clear that for t_{1}, t_{2} ∈ [0, 1], the hyperplane H_{t_{1},t_{2}} = {x ∈ ℝ^{4} : φ_{t_{1},t_{2}}(x) = 0} is a supporting hyperplane for S_{1} and that H_{t_{1},t_{2}} ∩ S_{1} = [L(t_{1}), L(t_{2})], so

(2.1) S_{1} − [L(t_{1}), L(t_{2})] is convex

For t_{1}, t_{2} ∈ [0, 1] let y_{t_{1},t_{2}} = (L(t_{1}) + L(t_{2}))/2. Define S = S_{1} − {y_{t_{1},t_{2}} : t_{1}, t_{2} ∈ ℚ ∩ [0, 1], t_{1} ≠ t_{2}} (2.2)

S is a G_{δ} set, since it is obtained by subtracting a countable set from a closed set. The set

\begin{matrix}S − ⋃\{[L(t_{1}), L(t_{2})] : t_{1}, t_{2} ∈ ℚ ∩ [0, 1]\} \\ = ⋂\{S_{1} − [L(t_{1}), L(t_{2})] : t_{1}, t_{2} ∈ ℚ ∩ [0, 1]\} \end{matrix} (2.3)

is convex by (2.1) and therefore the set

\begin{matrix}S = \\ S − ⋃\{[L(t_{1}), L(t_{2})] : t_{1}, t_{2} ∈ ℚ ∩ [0, 1]\} \\ ∪ ⋃_{t_{1},t_{2}∈ℚ∩[0,1]}[L(t_{1}), y_{t_{1},t_{2}}) \end{matrix} (2.4)

is countably convex.

Finally, put P = {L(t) : t ∈ ℚ ∩ [0, 1]}. From the construction it is clear that P ⊆ S, that P is dense in itself and that P is a 2-clique in S.

Theorem 5. If d ≤ 3 then a countably convex G_{δ} subset of ℝ^{d} does not contain a clique which is dense in itself.

Proof. By induction on d ≤ 3. If d = 0 then no subset of ℝ^{d} is dense in itself.

Suppose now that 0 < d ≤ 3 and, to the contrary, that S ⊆ ℝ^{d} is a G_{δ} set, S = ⋃_{n} C_{n} where each C_{n} is convex and that P ⊆ S has no isolated points and is a k-clique for some k ≥ 2.

Denote P̄ = cl_{S}P, the closure of P in S. Since P̄ is closed in S and S is G_{δ}, P̄ satisfies the Baire category theorem, and therefore for some n and an open ball u, it holds that u ∩ C_{n} ∩ P̄ is dense in u ∩ P̄. Denote C = C_{n} ∩ u. Thus C is convex and C ∩ P̄ is dense in P̄ ∩ u. Therefore, P ∩ u ⊆ cl_{S}C.

Let H be the affine span of C, and consider S′ = S∩H. This is again a countably convex G_{δ} set. Since H is closed, clC ⊆ H, so S ∩ H contains a dense in itself clique. If dim H < d, then a contradiction to the induction hypothesis is reached. Assume, then, that the affine span of C is ℝ^{d} (so C has nonempty interior in ℝ^{d}). The interior of C contains no points from P, so P is contained in the boundary of C.

If d = 1 then again a contradiction is reached, since the boundary of C is finite and cannot contain a dense in itself set. Suppose, then, that d > 1.

We argue next that P is contained in a finite union of hyperplanes in ℝ^{d}. Suppose this is not so. Then we can choose a sufficiently large subset {x_{0}, …, x_{ℓ−1}} ⊆ P in general position (and, since all points belong to ∂C, it will also be in convex position) and denote T = conv(x_{0}, …, x_{ℓ−1}). T is a convex polytope in ℝ^{d} whose vertices are {x_{0}, …, x_{ℓ−1}}. If d = 2 then T is a polygon and if d = 3 the T is a simplicial polytope, (that is, every face of T is a triangle). To contradict the assumption that P is a k-clique it is enough to find a k-element subset of vertices {y_{0}, …, y_{k−1}} so that conv(y_{0}, …, y_{k−1}) − {y_{0}, …, y_{k−1}} ⊆ int T.

If d = 2 then ℓ = 2k suffices, since for a polygon with 2k vertices, taking every second vertex in a cyclical ordering, gives a polygon that, except for its vertices is contained in the interior of the original polygon. If d = 3 then let ℓ = 4k. Since the graph of T is planar, the vertices of T are colorable by 4 colors so that no two vertices which are joined by an edge have the same color (by the 4-colors Theorem). Find k vertices with the same color, and observe that in a simplicial polytope in ℝ^{3}, two vertices which are not joined by an edge are joined by an inner diagonal.

Since P is contained in a finite union of hyperplanes, there exists a hyperplane F so that P ∩ F contains a dense in itself set. Now the induction hypothesis is violated by S ∩ F. This completes the proof.

Remarks: Our first remark is, that the full power of the 4-colors theorem is not really needed. All we need is some fixed bound on the chromatic number of all graphs of 3-polytopes — and a bound of 5 follows easily from Euler’s equation and is known since the middle of the 19-th century. Our second remark is that lower and upper bounds on the number of inner diagonals in a polytope were computed in [4].

We conclude this section with an example that shows that even in ℝ^{2} there are countably convex G_{δ} sets with nonempty radicals. It is easy to check that in ℝ^{1} no such example exists.

Example 6. There exists a G_{δ} set in ℝ^{2} which is countably convex and has a non-empty radical.

Proof. Let D be the closed unit disk in ℝ^{2} and let C be the upper half of its boundary. Subtract from F an open half-plane that contains exactly the middle third of C. Continue inductively, where at stage n, for each of the remaining 2^{n} sub arcs C_{i}^{n}, i < 2^{n} of C, an open half-plane which contains the middle third of C_{i}^{n} is subtracted from D. Let S_{n} be the set which is obtained at stage n and and let S = ⋂_{n} S_{n}.

Let P be the Cantor set constructed by this process inside C. Clearly, P ⊆ S. Put P = P_{1} ∪ P_{2} where P_{1} is the set of all end-points of all C_{i}^{n}, n < ∞, i < 2^{n} and P_{2} = P − P_{1}.

S is clearly a G_{δ} set, since S = int S ∪ P ∪ C′, where C′ is the lower half of ∂D: a union of an open set and two closed sets. Further, P is easily verified to be a semi-clique in S. Finally, S is countably convex, since int S ∪ P 0_{2} ∪ C′ is convex, while P_{1} is countable.

3. Infinite dimensional spaces

All Banach spaces in this Section are infinite-dimensional. We will see that the infinite-dimensional case is completely different from the finite-dimensional one. We start with two standard auxiliary results (which we prove just for the completeness of presentation) and then prove Theorem 9 which is the main ingredient for the main Corollary 12.

Recall that for a Banach space E, a subspace Y ⊆ E^{∗} is 1-norming if ||x|| = sup{f(x) : f ∈ B_{Y} } for each x ∈ E.

Lemma 7. A subspace Y ⊂ E^{∗} is 1-norming iff B_{Y} is w^{∗}-dense in B_{E^{∗}}.

Proof. Clearly, if B_{Y} is w^{∗}-dense in B_{E^{∗}} then Y ⊂ E^{∗} is 1-norming. Conversely, assume that Y ⊂ E^{∗} is 1-norming and suppose to the contrary that there is an f ∈ B_{E^{∗}} \ w^{∗}−clB_{Y}. Apply a separating theorem and find an x ∈ E such that f(x) > 1 > sup{g(x) : g ∈ w^{∗}−clB_{Y} }, a contradiction.

Lemma 8. Let E be a Banach space with separable dual. Then each norm-closed subset F of B_{E^{∗}} is a w^{∗}-G_{δ} set.

Proof. For each f ∈ B_{E^{∗}} \ F denote d_{f} = d(f, F). It is clear that

B_{E^{∗}} \ F = ∪_{f B_{E^{∗}} \F} ((f + 1/2intB_{E^{∗}}) ∩ B_{E^{∗}}).

By the Lindelöf theorem there is a countable subcovering, i.e.

B_{E^{∗}} \ F = ∪_{i=1}^{∞}((f_{i} + 1/2intB_{E^{∗}}) ∩ B_{E^{∗}}).

By passing to the closed balls we get

B_{E^{∗}} \ F = ∪_{i=1}^{∞}((f_{i} + 1/2B_{E^{∗}}) ∩ B_{E^{∗}}).

Since B_{E^{∗}} is w^{∗}-G_{δ} and each set (f_{i} + 1/2B_{E^{∗}}) ∩ B_{E^{∗}} is w^{∗}-closed, it follows that F is a w^{∗}-G_{δ} set.

By KerF, for F ∈ E^{∗}, the null space of the functional F is denoted.

Theorem 9. Let E be a Banach space with separable dual E^{∗} such that there is a 2-dimensional subspace M ⊂ E^{∗∗} with the property: for each F ∈ M, KerF is 1-norming. Then there is a sequence {C_{i}}_{i=1}^{∞} of convex subsets of B_{E^{∗}} such that

(i) The set S = ∪_{i=1}^{∞}C_{i} is w^{∗}-G_{δ}. (ii) Each C_{i} is w^{∗}-dense in S. (iii) There is a w^{∗}-dense in S subset P = {f_{i}} ⊂ S which does not contain w^{∗}-

isolated points and such that for i < j the interval [f_{i}, f_{j}] ⊄ S.

Proof. Let the functionals F, G ∈ S_{M} form a basis of M. Put

F_{1} = F, F_{n} = F + \frac{1}{n}G, n = 2, 3, ..., C_{n} = KerF_{n} ∩ B_{E^{∗}}, S = ∪_{i=1}^{∞}C_{i}.

To check that the set S is w^{∗}-G_{δ} in B_{E^{∗}} it is enough in view of Lemma 8, to show that S is norm-closed. Let lim h_{k} = h; h_{k} ∈ S. Since S = ∪_{i=1}^{∞}C_{i} and each C_{i} is norm-closed, we may assume that h_{k} ∈ C_{n_{k}}, n_{k} ≠ n_{m}, k ≠ m. However ||.||- lim F_{n_{k}} = F_{1} and hence 0 = F_{n_{k}}(h_{n_{k}}) → F_{1}(f). Thus h ∈ C_{1} ⊂ S which proves that S is ||.||-closed, and by Lemma 8, w^{∗}-G_{δ}.

By Lemma 7 each C_{n} is w^{∗}-dense in B_{E^{∗}}. Next put L = ∩_{n=1}^{∞}C_{n}. By using the definition of C_{n} it is easy to see that for each pair p ≠ q

L = C_{p} ∩ C_{q} (3.5)

For each n it is clear that C_{n} \L is norm-dense in C_{n} and thus C_{n} \L is w^{∗}-dense in B_{E^{∗}}.

We now construct the set P = {f_{i}} as in (iii). Fix a sequence {g_{i}} ⊂ B_{E^{∗}} which is w^{∗}-dense in B_{E^{∗}}. Denote by d a metric which generates the w^{∗}-topology on B_{E^{∗}}

Take f_{1} ∈ C_{1} \ L with d(g_{1}, f_{1}) < 1. Next, using the density of C_{2} \ L in B_{E^{∗}} find f_{2} ∈ C_{2} \ L with d(f_{1}, f_{2}) < 1.

Now choose f_{3} ∈ C_{3} \ L with d(g_{2}, f_{3}) < 1/2. Next we approximate f_{1}, f_{2}, f_{3}. Choose f_{4} ∈ C_{4} \ L, f_{5} ∈ C_{5} \ L, f_{6} ∈ C_{6} \ L so that

d(f_{1}, f_{4}) < 1/2, d(f_{2}, f_{5}) < 1/2, d(f_{3}, f_{6}) < 1/2

Now choose f_{7} ∈ C_{7} \ L with d(g_{3}, f_{7}) < 1/3. The rest of the construction is clear.

In this way we construct a dense in itself sequence P = {f_{i}}_{n=1}^{∞}, f_{i} ∈ C_{i} \ L such that each point in S (even in B_{E^{∗}}) is a w^{∗}-cluster point of P.

Check that for i < j the interval [f_{i}, f_{j}] is not contained in S. Indeed, from (3.5) it follows that for each p the intersection C_{p} ∩ [f_{i}, f_{j}] is either a singleton or empty. Thus the intersection S ∩ [f_{i}, f_{j}] is countable, which completes the proof of the theorem.

Remark 10. We shall show how to get an example of a Banach space E with the property required by Theorem 9. Let X be a Banach space with separable dual X^{∗} and such that dim X^{∗∗}/X = ∞, i.e. X is not quasi-reflexive (e.g. X = c_{0}). Let M ⊂ X^{∗∗} be a 2-dimensional subspace of X^{∗∗} with M ∩ X = {0}. Define a new norm in the space X as follows

|||x||| = sup{f(x) : f ∈ M_{⊥}, ||f|| = 1}

where M_{⊥} = {f ∈ l_{1} : F (f) = 0, F ∈ M}. We check now that the new norm is equivalent to the original one. First, from M ∩ X = {0} and dimM < ∞ it follows that

a = inf{d(x, M) : x ∈ S_{X}} > 0.

Take an x ∈ S_{X} and by the Hahn-Banach theorem find an f̃ ∈ S_{M^{⊥}} with f̃(x) = d(x, M) ≥ a. Now by using the w^{∗}-density of M_{⊥} in M^{⊥} we find for each δ > 0 a functional f_{δ} ∈ S_{M_{⊥}} so that |(f̃− f_{δ})(x)| < δ. Now it is clear that a||x|| ≤ |||x||| ≤ ||x|| for each x ∈ X.

It is trivial that the space E = (X, |||.|||) has the property required by Theorem 9.

Next we slightly change the formulation of Theorem 9. More precisely, we want to transfer the set S into some Banach space X in such a way that w^{∗}-properties of the sets S and P in E^{∗} coincide with the norm-properties in X.

Theorem 11. There exists a separable Banach space X which contains a subset S̃ with the following properties:

(i) K = clS̃ is compact. (ii) S̃ is a G_{δ} set. (iii) ˜ S̃ = ∪_{i=1}^{∞}C̃_{i} where each set C̃_{i} is convex and dense in S̃. (iv) There is a dense in itself 2-clique P̃ ⊂ S̃ which is dense in S.

Proof. Let {x_{i}}_{i=1}^{∞} be a dense subset of S_{E}. Define a weaker norm on E^{∗} as follows

|||f||| = ∑_{k=1}^{∞} 2^{−k}|f(x_{k})|.

Then the set K = (B_{E^{∗}}, |||.|||) is homeomorphic to (B_{E^{∗}}, w^{∗}) and, hence is a compact. In particular, (E^{∗}, |||.|||) = ∪_{n=1}^{∞}nB_{E^{∗}} is separable. Denote by X the completion of (E^{∗}, |||.|||). Thus X is a separable Banach space. Let J : E^{∗} → X be a natural embedding. Now use the notation of Theorem 9. Put C̃_{i} = J(C_{i}), S̃ = J(S) and P̃ = J(P). All the properties (i)-(iv) are clear.

Corollary 12. Every infinite dimensional Banach space Y contains subsets S and P which have all the properties of S̃ and P̃ from Theorem 11.

Proof. Let X be the Banach space from Theorem 11 and let T : X → Y be a compact 1-1 operator. Since K ⊂ X is compact, it follows that K_{1} = T (K) is a compact and hence, G_{δ}-set in Y. The restriction T ↾_{K} is an affine homeomorphism of K onto K_{1}. Clearly, the sets S = T (S̃) and P = T (P̃) have the same properties as the sets S̃ and P̃ from Theorem 11.

The authors wish to thank the referee for valuable remarks.

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Department of Mathematics, Ben Gurion University of the Negev