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 2d. 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 Lp norm

\ \|x\|_p=\left(|x_1|^p+|x_2|^p+\cdots+|x_d|^p\right)^{1/p}.

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

Normed vector spaces

Equilateral dimension has also been considered for normed vector spaces with norms other than the Lp 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 2d; 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 Lp 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\ge\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

This article is issued from Wikipedia - version of the Wednesday, August 19, 2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.