Equilateral dimension

In mathematics, the equilateral dimension of a metric space is the maximum number of points that are all at equal distances from each other.^[1] Equilateral dimension has also been called "metric dimension", but the term "metric dimension" also has many other inequivalent usages.^[1] The equilateral dimension of a d-dimensional Euclidean space is d + 1, and the equilateral dimension of a d-dimensional vector space with the Chebyshev distance (L^∞ norm) is 2^d. However, the equilateral dimension of a space with the Manhattan distance (L¹ norm) is not known; Kusner's conjecture, named after Robert B. Kusner, states that it is exactly 2d.^[2]

Lebesgue spaces

The equilateral dimension has been particularly studied for Lebesgue spaces, finite-dimensional normed vector spaces with the L^p norm

\ \|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\cdots +|x_{d}|^{p}\right)^{1/p}.

The equilateral dimension of L^p spaces of dimension d behaves differently depending on the value of p:

For p = 1, the L^p norm gives rise to Manhattan distance. In this case, it is possible to find 2d equidistant points, the vertices of an axis-aligned cross polytope. The equilateral dimension is known to be exactly 2d for d ≤ 4,^[3] and to be upper bounded by O(d log d) for any d.^[4] Robert B. Kusner suggested in 1983 that the equilateral dimension for this case should be exactly 2d;^[5] this suggestion (together with a related suggestion for the equilateral dimension when p > 2) has come to be known as Kusner's conjecture.
For 1 < p < 2, the equilateral dimension is at least (1 + ε)d where ε is a constant that depends on p.^[6]
For p = 2, the L^p norm is the familiar Euclidean distance. The equilateral dimension of d-dimensional Euclidean space is d + 1: the d + 1 vertices of an equilateral triangle, regular tetrahedron, or higher-dimensional regular simplex form an equilateral set, and every equilateral set must have this form.^[5]
For 2 < p < ∞, the equilateral dimension is at least d + 1: for instance the d basis vectors of the vector space together with another vector of the form (−x, −x, ...) for a suitable choice of x form an equilateral set. Kusner's conjecture states that in these cases the equilateral dimension is exactly d + 1. Kusner's conjecture has been proven for the special case that p = 4.^[6] When p is an odd integer the equilateral dimension is upper bounded by O(d log d).^[4]
For p = ∞ (the limiting case of the L^p norm for finite values of p, in the limit as p grows to infinity) the L^p norm becomes the Chebyshev distance, the maximum absolute value of the differences of the coordinates. For a d-dimensional vector space with the Chebyshev distance, the equilateral dimension is 2^d: the 2^d vertices of an axis-aligned hypercube are at equal distances from each other, and no larger equilateral set is possible.^[5]

Normed vector spaces

Equilateral dimension has also been considered for normed vector spaces with norms other than the L^p norms. The problem of determining the equilateral dimension for a given norm is closely related to the kissing number problem: the kissing number in a normed space is the maximum number of disjoint translates of a unit ball that can all touch a single central ball, whereas the equilateral dimension is the maximum number of disjoint translates that can all touch each other.

For a normed vector space of dimension d, the equilateral dimension is at most 2^d; that is, the L^∞ norm has the highest equilateral dimension among all normed spaces.^[7] Petty (1971) asked whether every normed vector space of dimension d has equilateral dimension at least d + 1, but this remains unknown. There exist normed spaces in any dimension for which certain sets of four equilateral points cannot be extended to any larger equilateral set^[7] but these spaces may have larger equilateral sets that do not include these four points. For norms that are sufficiently close in Banach–Mazur distance to an L^p norm, Petty's question has a positive answer: the equilateral dimension is at least d + 1.^[8]

It is not possible for high-dimensional spaces to have bounded equilateral dimension: for any integer k, all normed vector spaces of sufficiently high dimension have equilateral dimension at least k.^[9] more specifically, according to a variation of Dvoretzky's theorem by Alon & Milman (1983), every d-dimensional normed space has a k-dimensional subspace that is close either to a Euclidean space or to a Chebyshev space, where

k\geq \exp(c{\sqrt {\log d}})

for some constant c. Because it is close to a Lebesgue space, this subspace and therefore also the whole space contains an equilateral set of at least k + 1 points. Therefore, the same superlogarithmic dependence on d holds for the lower bound on the equilateral dimension of d-dimensional space.^[8]

Riemannian manifolds

For any d-dimensional Riemannian manifold the equilateral dimension is at least d + 1.^[5] For a d-dimensional sphere, the equilateral dimension is d + 2, the same as for a Euclidean space of one higher dimension into which the sphere can be embedded.^[5] At the same time as he posed Kusner's conjecture, Kusner asked whether there exist Riemannian metrics with bounded dimension as a manifold but arbitrarily high equilateral dimension.^[5]

Notes

References

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Swanepoel, Konrad J. (2004), "A problem of Kusner on equilateral sets", Archiv der Mathematik, 83 (2): 164–170, arXiv:math/0309317, doi:10.1007/s00013-003-4840-8, MR 2104945 .
Swanepoel, Konrad J.; Villa, Rafael (2008), "A lower bound for the equilateral number of normed spaces", Proceedings of the American Mathematical Society, 136 (1): 127–131, arXiv:math/0603614, doi:10.1090/S0002-9939-07-08916-2, MR 2350397 .

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