Large Equilateral Sets in Subspaces of ℓ∞n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _\infty ^n$$\end{document} of Small Codimension

For fixed k we prove exponential lower bounds on the equilateral number of subspaces of ℓ∞n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _{\infty }^n$$\end{document} of codimension k. In particular, we show that subspaces of codimension 2 of ℓ∞n+2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _{\infty }^{n+2}$$\end{document} and subspaces of codimension 3 of ℓ∞n+3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _{\infty }^{n+3}$$\end{document} have an equilateral set of cardinality n+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n+1$$\end{document} if n≥7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 7$$\end{document} and n≥12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 12$$\end{document} respectively. Moreover, the same is true for every normed space of dimension n, whose unit ball is a centrally symmetric polytope with at most 4n/3-o(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${4n}/{3}-o(n)$$\end{document} pairs of facets.


Introduction
The norm · ∞ of x ∈ R n is defined as x ∞ = max 1≤i≤n |x i |, and n ∞ denotes the normed space (R n , · ∞ ). In [5] Kobos studied subspaces of n ∞ of codimension 1, and proved the lower bound e(X ) ≥ 2 n/2 , which in particular implies Conjecture 1.1 for these spaces for n ≥ 6. In the same paper he proposed as a problem to prove Petty's conjecture for subspaces of n ∞ of codimension 2. In Theorem 1.2 we prove exponential lower bounds on the equilateral number of subspaces of n ∞ of codimension k. This, in particular, solves Kobos' problem if n ≥ 9. Theorem 1.2 Let X be an (n − k)-dimensional subspace of n ∞ . Then Note that none of the three bounds follows from the other two in Theorem 1.2, hence none of them is redundant. Comparing (1) and (3), for fixed k we have for some 0 < c k < 1, while 2 n−k /(n − k) k = 2 n−k−k log(n−k) = 2 n−o(n) . On the other hand, when we let k vary, it can be as large as (n) in (3) to still give a non-trivial estimate, while k can only be chosen up to O(n/ log n) for (1) to be non-trivial. Finally, (2) is beaten by (1) or (3) in most cases, however, for k = 2, 3 and for small values of n, (2) gives the best bound. Table 1 shows the lower bounds given by Theorem 1.2 for small values of k and n, whenever it is at least 3. For two n-dimensional normed spaces X , Y by d BM (X , Y ) = inf T { T · T −1 } we denote their Banach-Mazur distance, where the infimum is over all linear isomorphisms T : X → Y . The metric space of isometry classes of normed spaces endowed with the logarithm of the Banach-Mazur distance is the Banach-Mazur compactum. It is not hard to see that e(X ) is upper semi-continuous on the Banach-Mazur compactum. This, together with the fact that any convex polytope can be obtained as a section of a cube of sufficiently large dimension (see for example p. 72 of Grünbaum's book [4]) implies that it would be sufficient to prove Conjecture 1.1 for k-codimensional subspaces of n ∞ for all 1 ≤ k ≤ n − 4 and n ≥ 5. (This was also pointed out in [5].) Unfortunately, our bounds are only non-trivial if n is sufficiently large compared to k. However, we deduce an interesting corollary.
opposite pairs of facets. If X is a d-dimensional normed space with P as a unit ball, then e(X ) ≥ d + 1.
There have been some extensions of lower bounds obtained on the equilateral number of certain normed spaces to other norms that are close to them according to the Banach-Mazur distance. These results are based on using Brouwer's fixed point theorem, first applied in this context by Brass [2] and Dekster [3]. Swanepoel and Villa [10] proved that if d BM (Y , n ∞ ) ≤ 3/2, then e(Y ) ≥ n + 1. Kobos [5] proved that if X is a subspace of n ∞ of dimension n − 1 and Y is a normed space of dimension n − 1 such that d BM (X , Y ) ≤ 2, then e(Y ) ≥ n/2 . We prove a similar result for subspaces of n ∞ of codimension at least 1.

Norms with Polytopal Unit Ball and Small Codimension
We recall the following well-known fact to prove Corollary 1.3. (For a proof, see for example [1].) and apply inequality (3) from Theorem 1.2. This yields e(X ) ≥ d + 1.
To confirm Petty's conjecture for subspaces of n ∞ of codimension 2 and 3 when n ≥ 9 and respectively n ≥ 15, apply inequality (2) from Theorem 1.2 with = 2.

Notation
We denote vectors by bold lowercase letters, and the i-th coordinate of a vector a ∈ R n by a i . We treat vectors by default as column vectors. By subspace we mean linear subspace. We write span (a 1 , . . . , a k ) for the subspace spanned by a 1 , . . . , a k ∈ R n . For a subspace X ⊆ R n we denote by X ⊥ the orthogonal complement of X . We denote by

Idea of the Constructions
For two vectors x, y ∈ X we have x − y ∞ = c if and only if the following hold: In our constructions of c-equilateral sets S ⊆ X , we split the index set [n] of the coordinates into two parts, [n] = N 1 ∪ N 2 . In the first part N 1 , we choose all the coordinates from the set {0, 1, −1}, so that for each pair from S there will be an index in N 1 for which (4) holds, and (5) is not violated by any index in N 1 . We use N 2 to ensure that all of the points we choose are indeed in the subspace X . For each vector, this will lead to a system of linear equations. The main difficulty will be to choose the values of the coordinates in N 1 so that the coordinates in N 2 , obtained as a solution to those systems of linear equations, do not violate (5).

Proof of Theorem 1.2
we denote by A| N the submatrix of A formed by those columns of A whose index is in N . Similarly, for a vector v ∈ R n we denote by v| N the vector that is formed by those coordinates of v whose index is in N (without changing the order of the indices). (1) We construct a 2-equilateral set of size 2 n−k /(n − k) k . Let B be a k × k submatrix of A for which |det B| is maximal among all k × k submatrices. Further, let N 2 be the set of indices of the columns of B and N 1 = [n] \ N 2 . Note that det B = 0, since the vectors {a i : i ∈ [k]} are linearly independent. First, we find many vectors in X whose coordinates in N 1 are from {1, −1}, and then we select a subset of these that form an equilateral set. For every J ⊆ N 1 we define the vector w(J ) ∈ R n as

Proof of
To see that w(J ) is in X , we need to check that Aw(J ) = 0. This is indeed the case, since by Cramer's rule w(J )| N 2 is the solution of Further, by the multilinearity of the determinant for every J and j we have By the maximality of |det B| and by the triangle inequality we have This implies that for each J and However, we can find a 2-equilateral subset of W that has large cardinality, as follows. Proof of (2) Fix some 1 ≤ ≤ n/(k + 1). We will construct a 1-equilateral set of cardinality Let I 1 , . . . , I k ⊆ [n] be sets of cardinality at most , and σ ∈ {±1} n be a sign vector with the following properties: Note that det B > 0, since the vectors a 1 , . . . , a k are linearly independent. Let N 2 = i∈[k] I i and N 1 = [n] \ N 2 . We find a set of vectors whose coordinates in N 1 are from {0, 1, −1}, and then we find a large subset of them that forms an equilateral set.
For every subset J ⊆ N 1 of cardinality at most we define the vector w(J ) ∈ R n as To see that w(J ) is in X , we have to check that Aw(J ) = 0. This follows by showing that w(J )| N 2 is a solution of By Cramer's rule y ∈ R k defined as is a solution of By = b J ,σ . Thus, w(J )| N 2 is indeed a solution of (6) of a special form where x i = σ i y j if i ∈ I j . Note that B( j, b J ,σ ) = B(b J 1 ,σ , . . . , b J k ,σ ) for some disjoint sets J 1 , . . . , J k , hence by property ( * ) we have Further, the multilinearity of the determinant together with property ( * * ) implies that for i ∈ I k we have   (7), by the definition of W (s), and by (8). Finally, it is not hard to see that we can add 0 to W (s). Thus W (s) ∪ {0} is a 1-equilateral set of the promised cardinality.
In the proof of (3) we will need the following simple lemma.

Lemma 3.1 Let A be a real matrix of size k × n. Then for every ε > 0 there exists a real matrix A of size k × n such that |a i j − a i j | ≤ ε for all i, j, and every k × k minor of A is non-zero.
Proof Associating matrices with points of R n×k , the set of those matrices in which a given k × k minor is 0 is the zero set of a non-zero polynomial, which is nowhere dense. Thus, the set of those matrices for which there is a k × k minor which is zero is nowhere dense, which implies the statement. Proof of (3) Fix some 1 ≤ ≤ n/(2k + 1). We will construct a 1-equilateral set of cardinality 1≤r ≤ n − 2k r + 1.
Pick 2 disjoint submatrices B 1 , . . . , B 2 of A of size k × k, such that for every 1 ≤ m ≤ k and 1 ≤ i ≤ 2 and for any column b r that is not a column of any B j we have This we may do by choosing the B i 's after each other, always choosing the submatrix with the largest determinant disjoint from the previous submatrices. Using Lemma 3.1 and the upper semi-continuity of the equilateral number, we may further assume that Let U i be the set of indices of the columns of B i , let N 2 = U 1 ∪ · · · ∪ U 2 and N 1 = [n] \ N 2 . We will find vectors in X (denoted by y(J )) whose coordinates in N 1 are from the set {0, −1}, and whose coordinates in N 2 have absolute value at most 1/2. We do not construct them directly, but as the sum of some other vectors w(J , i), z(J , i) ∈ X , whose coordinates in N 1 are from {0, −1/2}.
For every set J ={q 1 , . . . , q |J | }⊆[N 1 ] of cardinality at most with q 1 < · · · < q |J | , and for every 1 ≤ i ≤ |J | we define w(J , i) ∈ R n and z(J , i) ∈ R n as To see that w(J , i) and z(J , i) are in X , we need to check that Aw(J , i) = 0 and Az(J , i) = 0. This is indeed the case, since by Cramer's rule both w(J , i)| U 2i and z(J , i)| U 2i−1 are solutions of Therefore y(J ) = 1≤i≤|J | (w(J , i) + z(J , i)) is also in X . Note that by assumption (9) and by the multilinearity of the determinant we have |w(J , i) j |, |z(J , i) j | ≤ 1/2 for all 1 ≤ j ≤ n. For the coordinates of y(J ) we have Thus, for any two distinct sets J 1 , J 2 ⊆ [N 1 ] of cardinality at most there is a coordinate j ∈ [N 1 ] for which {y(J 1 ) j , y(J 2 ) j } = {0, −1}, and for all 1 ≤ j ≤ n we have |y(J 1 ) j − y(J 2 ) j | ≤ 1. This means y(J 1 ) − y(J 2 ) ∞ = 1, and

Equilateral Sets in Normed Spaces Close to Subspaces of n ∞
In this section we prove Theorem 1.4. The construction we use is similar to the one from [10]. Fix some 1 ≤ ≤ (n − 2k)/k, and let N = n − k(2 + ) and c = /(2(N − 1)) > 0. We assume that the linear structure of Y is identified with the linear structure of X and the norm · Y of Y satisfies Conditions (10)-(13) imply that p s (ε)−p t (ε) ∞ = 1+ε s t for every 1 ≤ s < t ≤ N . To finish the proof, we only have to find vectors p j (ε) that satisfy conditions (10)-(13). We construct them in a similar way as the equilateral sets in the proof of Theorem 1.2. Select 2 + disjoint submatrices B 1 , . . . , B 2 of A of size k × k, such that for every 1 ≤ m ≤ k and 1 ≤ i ≤ + 2, and for any column b r that is not a column of any B j , we have This we may do by choosing the B i 's after each other, always choosing the submatrix with the largest determinant disjoint from the previous submatrices. Using Lemma 3.1 and the upper semi-continuity of the equilateral number, we may further assume that |det B i | > 0 for all 1 ≤ i ≤ + 2. Let U i be the set of indices of the columns of B i , and let N 2 = U 1 ∪ · · · ∪ U +2 . By permuting the coordinates, we may assume that N 2 ∩ [N ] = ∅. Indeed, permuting the coordinates gives a subspace that is linearly isometric to the initial one, and the equilateral number is the same for isometric normed spaces. We construct p j (ε) as a sum of 2 + other vectors p j (ε,1),. . ., p j (ε, 2 + ). For 1 ≤ j ≤ N we define p j (ε, q) as follows. For q ∈ {1, 2} let Let s(ε, j) = r < j ε r j b r / and for q ∈ {3, . . . , 2 + } let As before, by Cramer's rule we obtain that p j (ε, q) is contained in Y for every q and j, thus p j (ε) = q∈[2+ ] p j (ε, q) is also contained in Y . It follows immediately that p j (ε) satisfies conditions (10)-(12). It only remains to check condition (13).
By the multilinearity of the determinant, (14), and the triangle inequality, for every j, for q ∈ {1, 2} and i ∈ U q we have For every j, for q ∈ {3, . . . , 2 + } and i ∈ U q we have The above bounds on | p j (ε, q) i | imply that condition (13) holds for p j (ε), finishing the proof.

Concluding Remarks
In Theorem 1.4 we did not make an attempt to find the most optimal bound on the equilateral number in terms of the Banach-Mazur distance that can be achieved with this approach. Our goal with this theorem was to illustrate how to find non-trivial lower bounds on the equilateral number depending on the codimension and the Banach-Mazur distance, while keeping the proof short. One possible immediate improvement is to increase the lower bound by 1 without changing the bound on the Banach-Mazur distance. This can be done by adding a vector to the construction of the form p N +1 (ε) = (ε 1 N +1 , . . . , ε N N +1 , 0, . . . , 0, ≤ 1/2, . . . , ≤ 1/2). As this improvement is negligible for large k, we decided not to include it in the proof to keep it simpler.
With this plus one improvement, the lower bound on e(Y ) for k = 0 would match the bound from [10]. However, the bound on the Banach-Mazur distance in the assumption would be weaker. To explain this, note that our proof consists of two parts. In the first part we find possible candidates for points of a large equilateral set. This part is very similar to the proof in [10]. In the second part we have to fit these candidates in a subspace, and with doing this we lose some flexibility with the Banach-Mazur distance. For k = 0, however, we would not have a second part, thus the proof would be exactly the same as in [10].
Note also that for k = 1, = n/2 + 1 our lower bound in Theorem 1.4 is the same as the bound in [5], but again with a worse bound on the Banach-Mazur distance. The reason for this is that our intention was to present a non-trivial general bound that can be proven in a short way in various settings of the parameters.
Finally, it would be interesting to find some applications of Theorem 1.4 (or of a more optimized form of it). One corollary which was pointed out by an anonymous referee is that if Y is a subspace of n p of codimension k for p ≥ 2, then e(Y ) ≥ k, p (n 1−1/ p ), since d BM ( n ∞ , n p ) = n 1/ p .