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. Equilateral dimension has also been called "metric dimension", but the term "metric dimension" also has many other inequivalent usages. The equilateral dimension of a formula_1-dimensional Euclidean space is formula_2, achieved by a regular simplex, and the equilateral dimension of a formula_1-dimensional vector space with the Chebyshev distance (formula_4 norm) is formula_5, achieved by a hypercube. However, the equilateral dimension of a space with the Manhattan distance (formula_6 norm) is not known; Kusner's conjecture, named after Robert B. Kusner, states that it is exactly formula_7, achieved by a cross polytope.
Lebesgue spaces.
The equilateral dimension has been particularly studied for Lebesgue spaces, finite-dimensional normed vector spaces with the formula_8 norm
formula_9
The equilateral dimension of formula_8 spaces of dimension formula_1 behaves differently depending on the value of formula_12:
Normed vector spaces.
Equilateral dimension has also been considered for normed vector spaces with norms other than the formula_8 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 formula_1, the equilateral dimension is at most formula_5; that is, the formula_4 norm has the highest equilateral dimension among all normed spaces. asked whether every normed vector space of dimension formula_1 has equilateral dimension at least formula_2, 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 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 formula_8 norm, Petty's question has a positive answer: the equilateral dimension is at least formula_2.
It is not possible for high-dimensional spaces to have bounded equilateral dimension: for any integer formula_54, all normed vector spaces of sufficiently high dimension have equilateral dimension at least formula_54. more specifically, according to a variation of Dvoretzky's theorem by , every formula_1-dimensional normed space has a formula_54-dimensional subspace that is close either to a Euclidean space or to a Chebyshev space, where
formula_58
for some constant formula_59. Because it is close to a Lebesgue space, this subspace and therefore also the whole space contains an equilateral set of at least formula_60 points. Therefore, the same superlogarithmic dependence on formula_1 holds for the lower bound on the equilateral dimension of formula_1-dimensional space.
Riemannian manifolds.
For any formula_1-dimensional Riemannian manifold the equilateral dimension is at least formula_2. For a formula_1-dimensional sphere, the equilateral dimension is formula_66, the same as for a Euclidean space of one higher dimension into which the sphere can be embedded. 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.