Equilateral dimension

In mathematics, the equilateral dimension of a metric space is the maximum size of any subset of the space whose points are all at equal distances to each other.

[1] Equilateral dimension has also been called "metric dimension", but the term "metric dimension" also has many other inequivalent usages.

, achieved by the vertices of a regular simplex, and the equilateral dimension of a

-dimensional vector space with the Chebyshev distance (

, achieved by the vertices of a hypercube.

However, the equilateral dimension of a space with the Manhattan distance (

Kusner's conjecture, named after Robert B. Kusner, states that it is exactly

, achieved by the vertices of a cross polytope.

[2] The equilateral dimension has been particularly studied for Lebesgue spaces, finite-dimensional normed vector spaces with the

behaves differently depending on the value of

Equilateral dimension has also been considered for normed vector spaces with norms other than the

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

norm has the highest equilateral dimension among all normed spaces.

[7] Petty (1971) asked whether every normed vector space of dimension

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

norm, Petty's question has a positive answer: the equilateral dimension is at least

[8] It is not possible for high-dimensional spaces to have bounded equilateral dimension: for any integer

, all normed vector spaces of sufficiently high dimension have equilateral dimension at least

[9] more specifically, according to a variation of Dvoretzky's theorem by Alon & Milman (1983), every

-dimensional subspace that is close either to a Euclidean space or to a Chebyshev space, where

Because it is close to a Lebesgue space, this subspace and therefore also the whole space contains an equilateral set of at least

Therefore, the same superlogarithmic dependence on

holds for the lower bound on the equilateral dimension of

-dimensional Riemannian manifold the equilateral dimension is at least

-dimensional sphere, the equilateral dimension is

, 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.