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.^[1] The equilateral dimension of ${\displaystyle d}$-dimensional Euclidean space is ${\displaystyle d+1}$, achieved by a regular simplex, and the equilateral dimension of a ${\displaystyle d}$-dimensional vector space with the Chebyshev distance (${\displaystyle L^{\infty }}$ norm) is ${\displaystyle 2^{d}}$, achieved by a hypercube. However, the equilateral dimension of a space with the Manhattan distance (${\displaystyle L^{1}}$ norm) is not known. Kusner's conjecture, named after Robert B. Kusner, states that it is exactly ${\displaystyle 2d}$, achieved by a cross polytope.^[2]

Lebesgue spaces

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

The equilateral dimension of ${\displaystyle L^{p}}$ spaces of dimension ${\displaystyle d}$ behaves differently depending on the value of ${\displaystyle p}$:

• For ${\displaystyle p=1}$, the ${\displaystyle L^{p}}$ norm gives rise to Manhattan distance. In this case, it is possible to find ${\displaystyle 2d}$ equidistant points, the vertices of an axis-aligned cross polytope. The equilateral dimension is known to be exactly ${\displaystyle 2d}$ for ${\displaystyle d\leq 4}$,^[3] and to be upper bounded by ${\displaystyle O(d\log d)}$ for all ${\displaystyle d}$.^[4] Robert B. Kusner suggested in 1983 that the equilateral dimension for this case should be exactly ${\displaystyle 2d}$;^[5] this suggestion (together with a related suggestion for the equilateral dimension when ${\displaystyle p>2}$) has come to be known as Kusner's conjecture.
• For ${\displaystyle 1<p<2}$, the equilateral dimension is at least ${\displaystyle (1+\varepsilon )d}$ where ${\displaystyle \varepsilon }$ is a constant that depends on ${\displaystyle p}$.^[6]
• For ${\displaystyle p=2}$, the ${\displaystyle L^{p}}$ norm is the familiar Euclidean distance. The equilateral dimension of ${\displaystyle d}$-dimensional Euclidean space is ${\displaystyle d+1}$: the ${\displaystyle d+1}$ 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 ${\displaystyle 2<p<\infty }$, the equilateral dimension is at least ${\displaystyle d+1}$: for instance the ${\displaystyle d}$ basis vectors of the vector space together with another vector of the form ${\displaystyle (-x,-x,\dots )}$ for a suitable choice of ${\displaystyle x}$ form an equilateral set. Kusner's conjecture states that in these cases the equilateral dimension is exactly ${\displaystyle d+1}$. Kusner's conjecture has been proven for the special case that ${\displaystyle p=4}$.^[6] When ${\displaystyle p}$ is an odd integer the equilateral dimension is upper bounded by ${\displaystyle O(d\log d)}$.^[4]
• For ${\displaystyle p=\infty }$ (the limiting case of the ${\displaystyle L^{p}}$ norm for finite values of ${\displaystyle p}$, in the limit as ${\displaystyle p}$ grows to infinity) the ${\displaystyle L^{p}}$ norm becomes the Chebyshev distance, the maximum absolute value of the differences of the coordinates. For a ${\displaystyle d}$-dimensional vector space with the Chebyshev distance, the equilateral dimension is ${\displaystyle 2^{d}}$: the ${\displaystyle 2^{d}}$ 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 ${\displaystyle L^{p}}$ 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 ${\displaystyle d}$, the equilateral dimension is at most ${\displaystyle 2^{d}}$; that is, the ${\displaystyle L^{\infty }}$ norm has the highest equilateral dimension among all normed spaces.^[7] Petty (1971) asked whether every normed vector space of dimension ${\displaystyle d}$ has equilateral dimension at least ${\displaystyle d+1}$, 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 ${\displaystyle L^{p}}$ norm, Petty's question has a positive answer: the equilateral dimension is at least ${\displaystyle d+1}$.^[8]

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

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

Riemannian manifolds

For any ${\displaystyle d}$-dimensional Riemannian manifold the equilateral dimension is at least ${\displaystyle d+1}$.^[5] For a ${\displaystyle d}$-dimensional sphere, the equilateral dimension is ${\displaystyle d+2}$, 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