Equilateral dimension

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

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

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]

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

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

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]

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]