Positive bases, cones, Helly-type theorems

1 Introduction and main result

This paper is about Helly-type properties of families of convex sets. Suppose for instance that 𝓒 is a finite family of convex sets in ℝ^d and ⋂ 𝓒 (the intersection of the sets in 𝓒) contains a halfline. Then of course ⋂ 𝓒^* also contains a halfline for every subfamily 𝓒^* of 𝓒. In the opposite direction assume that ⋂ 𝓒^* contains a halfline for every subfamily 𝓒^* ⊂ 𝓒 of size at most m. Does it follow that ⋂ 𝓒 contains a halfline if m = m(d) is chosen suitably? The answer is yes according to a theorem of Katchalski [6].

A halfline is a one-dimensional cone. More generally a cone K with apex u ∈ ℝ^d and generator set V ⊂ ℝ^d is the set of points of the form u+∑v∈Wnα(v)v where W ⊂ V is finite and α(v) ≥ 0 for every v ∈ W. So K consists of all finite and positive combinations of elements of V translated by the vector u. The cone K is polyhedral if V is finite. For properties of cones, their lineality spaces, their polar cones, etc., see for instance Gruber’s book [4] or Schneider’s [10]. Our theorems are stated in Sections 1 and 2, and their proofs given in Sections 3, 4, and 5.

Our first result extends Katchalski’s theorem to k-dimensional cones where k ∈ [d] = {1, …, d}. Set m(k, d) = max{d + 1, 2(d – k + 1)}.

Note that the case k = 0 (which is not covered here) is Helly’s theorem and we could have stated it by defining m(0, d) = d + 1.

The following examples show that the value of m(k, d) is optimal. We write ab for the scalar product of vectors a, b ∈ ℝ^d.

In these examples the convex sets in the family 𝓒 are halfspaces of the form {x ∈ ℝ^d : ax ≤ 0}. This is no coincidence: the proof of Theorem 1.2 begins with a reduction from the family 𝓒 to a family of halfspaces of this form. For the case of such halfspaces, duality (or polarity) leads to a Helly-type theorem on the dimension of the lineality space of cones (Theorem 2.1), which is the other main result in this paper.

We remark that the long (and old and neat) survey paper by Danzer, Grünbaum, and Klee [2] contains several similar Helly-type results. The same applies to the more recent books by Gruber [4], Matoušek [8], and Schneider [10]. An up-to-date survey is by Bárány and Kalai [1].

2 A Helly-type theorem for the lineality space of cones

Suppose A ⊂ ℝ^d is a finite set and define pos A as the cone hull of A = {a₁, …, a_n}, i.e., pos A = { ∑1n α_ia_i : α_i ≥ 0 for all i ∈ [n]}. Analogously, lin A denotes the linear hull of A ⊂ ℝ^d which is the set of all linear combinations of the elements of A. The lineality space of pos A, to be denoted by lpos A, is the set pos A ∩ (–pos A) which is a linear subspace of ℝ^d, actually the (unique) maximal dimension subspace that pos A contains, see for instance [4]. Set h(k, d) = max{d + 1, 2(k + 1)}.

This is a Helly-type result for the dimension of the lineality space. Its proof is given in the next section. The value of h(k, d) is optimal again as the following examples show. Define A = {v₁, …, v_d+1} with the v_i coming from Example 1, then lpos A = ℝ^d while dim lpos B = 0 for all B ⊂ A, |B| ≤ d. This shows that h(k, d) is optimal when d + 1 ≥ 2(k + 1). For d + 1 ≤ 2(k + 1) let A_k be the set of vectors {± e₁, …, ± e_k}. Now dim lpos A_k = k but dim lpos A_k ∖ {a} < k for every a ∈ A_k. These examples indicate a duality between Theorems 1.2 and 2.1 that will become more transparent later.

The proof method of Theorem 1.2 works in other cases. For instance a theorem of Katchalski [5] states the following.

The case k = 0 is exactly Helly’s theorem. We mention further that, from this result, Theorem 1.2 (and in particular Theorem 1.1) can be deduced in a few lines. This is explained in Section 5. Another example is the following result of de Santis [3].

Examples 1 and 2 show that the bounds m(k, d) (in Theorem 2.2) and k + 1 (in Theorem 2.3) are optimal. Section 5 explains how the proof method of Theorem 1.2 applies to the last two results.

3 Proof of Theorem 2.1

Define L = lpos A and choose a positive basis X ⊂ A of L. This is just a subset of A with pos X = L but pos (X ∖ {x}) ≠ L for any x ∈ X. We are going to use a theorem of Reay [9] stating that a positive basis X has a partition X₁ ∪ … ∪ X_r such that |X₁| ≥ |X₂| ≥ … ≥ |X_r| ≥ 2 and X₁ ∪ … ∪ X_j is a positive basis for lin (X₁ ∪ … ∪ X_j) whose dimension is |X₁ ∪ … ∪ X_j| – j for every j ∈ [r]. Note that this and |X_i| ≥ 2 imply that dim lin (X₁ ∪ … ∪ X_j) ≥ j.

We show first that, in our case, |X₁| ≤ k + 1. As X₁ is a positive basis of lin X₁ whose dimension is |X₁| – 1 ≤ d we have |X₁| ≤ d + 1 ≤ h(k, d). Then, according to the condition of the theorem, |X₁| – 1 = dim pos X₁ ≤ k and |X₁| ≤ k + 1 indeed.

Set B_j = X₁ ∪ … ∪ X_j. We show, by induction on j, that |B_j| ≤ k + j or, equivalently that dim lin B_j ≤ k. The starting step j = 1 was just fixed so we move to step j – 1 → j assuming that j ≥ 2. Since X_j has the smallest size among the sets X₁, …, X_j we have |Xj|≤1j−1|Bj−1|. By the induction hypothesis |B_j–1| ≤ k + (j – 1). As we have checked above j ≤ dim lin (X₁ ∪ … ∪ X_j) = dim lin B_j. Thus

and j≤jkj−1, implying that j – 1 ≤ k.

This claim implies that |B_j| ≤ h(d, k) and then the condition of Theorem 2.1 gives dim lin B_j ≤ k. Consequently |B_j| ≤ k + j, completing the induction.

For j = r, Claim 3.1 gives dim L = dim lin B_r = dim lpos A ≤ k.□

4 Proof of Theorem 1.2

As m(k, d) ≤ d + 1, Helly’s theorem shows that ⋂ 𝓒 is non-empty. We assume, after a translation if necessary, that 0 ∈ ⋂ 𝓒. We assume further that every C ∈ 𝓒 is closed. This assumption is justified since a convex set contains a cone if and only if its closure contains a translate of this cone.

We agree that from now on the word “cone” means a cone with apex at the origin. In view of Lemma 4.1, 𝓒 satisfies the condition

After these preparations the proof starts by reducing or changing the family 𝓒 in two steps. For the first step we choose a k-dimensional cone K(𝓒^*) ⊂ ⋂ 𝓒^* for every 𝓒^* ⊂ 𝓒 satisfying |𝓒^*| ≤ m(k, d), and we choose K(C^*) so that it has exactly k (of course linearly independent) generators. Replace each C ∈ 𝓒 by the cone hull (or what is the same, convex hull), D(C), of the union of all cones K(𝓒^*) with C ∈ 𝓒^* and set 𝓓 = {D(C) : C ∈ 𝓒}. The new system 𝓓 consists of polyhedral cones and satisfies condition (*). Moreover ⋂ 𝓓 ⊂ ⋂ 𝓒 because D(C) ⊂ C. So it suffices to show that ⋂ 𝓓 contains a k-dimensional cone.

For the second reduction we observe that each D(C) is the intersection of finitely many closed halfspaces, H, of the form {x ∈ ℝ^d : ax ≤ 0} for some a ∈ ℝ^d (a ≠ 0), the outer normal of H. Let 𝓗 be the collection of these (finitely many) closed halfspaces, and let A ⊂ ℝ^d be the set of the corresponding outer normals. Evidently 𝓗 satisfies condition (*) and ⋂ 𝓒 contains a k-dimensional cone if so does ⋂ 𝓗 = ⋂ 𝓓.

The solution set of the system of linear inequalities

coincides with ⋂ 𝓗 and then ⋂ 𝓗 is the polar of the cone K = pos A. Let L be the lineality space of K, then K is the sum of L and the cone L^⊥ ∩ pos A, see [4: Proposition 3.3]. The latter cone is a pointed cone in ℝ^d and in the subspace L^⊥ as well. It coincides with the cone hull of the orthogonal projection, to be denoted by pr A, of A onto L^⊥. Using standard properties of polarity (c.f. [4]) we see that

where the polars are taken in ℝ^d. Thus ⋂ 𝓗 is a cone in L^⊥ and is full dimensional there because L^⊥ ∩ pos A = pos pr A is a pointed cone. Then ⋂ 𝓗 contains a k-dimensional cone if and only if dim L^⊥ = dim (lpos A)^⊥ is at least k.

Condition (*) for 𝓗 says that ⋂ 𝓗^* contains a k-dimensional cone for every 𝓗^* ⊂ 𝓗 with |𝓗^*| ≤ m(k, d). Writing B ⊂ A for the outer normals of the halfspaces in 𝓗^* we have ⋂ 𝓗^* = (pos B)^∘. The previous argument applies to B (instead of A) and gives that ⋂ 𝓗^* contains a k-dimensional cone if and only if dim (lpos B)^⊥ ≥ k.

Then for every B ⊂ A with |B| ≤ m(k, d), we have dim (lpos B)^⊥ ≥ k, or, what is the same, dim lpos B ≤ d – k. Theorem 2.1 applies now and shows that dim lpos A ≤ d – k, and equivalently dim L^⊥ ≥ k. Thus ⋂ 𝓗 contains a k-dimensional cone.□

A byproduct of this proof is an interesting corollary for systems of homogeneous inequalities.

5 Proofs of Theorems 2.2 and 2.3

We explain next how Katchalski’s theorem (Theorem 2.2) implies Theorem 2.1. We assume again that every C ∈ 𝓒 is closed and contains the origin. The same reduction as in the proof of Theorem 2.1 works again: for each 𝓒^* ⊂ 𝓒 whose size is at most m(k, d) we choose the same cone K(C^*) ⊂ ⋂ 𝓒^* whose generators are k linearly independent vectors from ⋂ 𝓒^*. Replace each C ∈ 𝓒 by the cone hull, D(C), of the union of all cones K(𝓒^*) with C ∈ 𝓒^* and set 𝓓 = {D(C) : C ∈ 𝓒}. As we have seen D(C) ⊂ C for every C ∈ 𝓒. Then dim ⋂ 𝓓^* ≥ k for every 𝓓^* ⊂ 𝓓 whose size is at most m(k, d). Thus Theorem 2.2 implies that dim ⋂ 𝓓 ≥ k and then there are linearly independent vectors v₁, …, v_k ∈ ⋂ 𝓓 ⊂ ⋂ 𝓒. The cone hull of these vectors is a cone of dimension at least k in ⋂ 𝓒, finishing the proof.

There is a related result by Kołodziejczyk [7] stating a condition, similar to (*), guaranteeing that ⋂ 𝓒 contains an k-dimensional affine halfspace. The condition is that ⋂ 𝓒^* contains an k-dimensional affine halfspace for every 𝓒^* ⊂ 𝓒 whose size is at most m(k, d). This theorem can also be proved by the same method.