Английская Википедия: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 Шаблон:Nowrap Euclidean space is <math>d+1</math>, achieved by a regular simplex, and the equilateral dimension of a Шаблон:Nowrap vector space with the Chebyshev distance (<math>L^\infty</math> norm) is <math>2^d</math>, achieved by a hypercube. However, the equilateral dimension of a space with the Manhattan distance (<math>L^1</math> norm) is not known. Kusner's conjecture, named after Robert B. Kusner, states that it is exactly <math>2d</math>, 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 <math>L^p</math> norm <math display=block>\ \|x\|_p=\bigl(|x_1|^p+|x_2|^p+\cdots+|x_d|^p\bigr)^{1/p}.</math>
The equilateral dimension of <math>L^p</math> spaces of dimension <math>d</math> behaves differently depending on the value of <math>p</math>: Шаблон:Unsolved
- For <math>p=1</math>, the <math>L^p</math> norm gives rise to Manhattan distance. In this case, it is possible to find <math>2d</math> equidistant points, the vertices of an axis-aligned cross polytope. The equilateral dimension is known to be exactly <math>2d</math> for <math>d\le 4</math>,[3] and to be upper bounded by <math>O(d\log d)</math> for all <math>d</math>.[4] Robert B. Kusner suggested in 1983 that the equilateral dimension for this case should be exactly <math>2d</math>;[5] this suggestion (together with a related suggestion for the equilateral dimension when <math>p>2</math>) has come to be known as Kusner's conjecture.
- For <math>1<p<2</math>, the equilateral dimension is at least <math>(1+\varepsilon)d</math> where <math>\varepsilon</math> is a constant that depends on <math>p</math>.[6]
- For <math>p=2</math>, the <math>L^p</math> norm is the familiar Euclidean distance. The equilateral dimension of <math>d</math>-dimensional Euclidean space is <math>d+1</math>: the <math>d+1</math> 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 <math>2<p<\infty</math>, the equilateral dimension is at least <math>d+1</math>: for instance the <math>d</math> basis vectors of the vector space together with another vector of the form <math>(-x,-x,\dots)</math> for a suitable choice of <math>x</math> form an equilateral set. Kusner's conjecture states that in these cases the equilateral dimension is exactly <math>d+1</math>. Kusner's conjecture has been proven for the special case that <math>p=4</math>.[6] When <math>p</math> is an odd integer the equilateral dimension is upper bounded by <math>O(d\log d)</math>.[4]
- For <math>p=\infty</math> (the limiting case of the <math>L^p</math> norm for finite values of <math>p</math>, in the limit as <math>p</math> grows to infinity) the <math>L^p</math> norm becomes the Chebyshev distance, the maximum absolute value of the differences of the coordinates. For a <math>d</math>-dimensional vector space with the Chebyshev distance, the equilateral dimension is <math>2^d</math>: the <math>2^d</math> 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 <math>L^p</math> 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 <math>d</math>, the equilateral dimension is at most <math>2^d</math>; that is, the <math>L^\infty</math> norm has the highest equilateral dimension among all normed spaces.[7] Шаблон:Harvtxt asked whether every normed vector space of dimension <math>d</math> has equilateral dimension at least <math>d+1</math>, 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 <math>L^p</math> norm, Petty's question has a positive answer: the equilateral dimension is at least <math>d+1</math>.[8]
It is not possible for high-dimensional spaces to have bounded equilateral dimension: for any integer <math>k</math>, all normed vector spaces of sufficiently high dimension have equilateral dimension at least <math>k</math>.[9] more specifically, according to a variation of Dvoretzky's theorem by Шаблон:Harvtxt, every <math>d</math>-dimensional normed space has a <math>k</math>-dimensional subspace that is close either to a Euclidean space or to a Chebyshev space, where <math display=block>k\ge\exp(c\sqrt{\log d})</math> for some constant <math>c</math>. Because it is close to a Lebesgue space, this subspace and therefore also the whole space contains an equilateral set of at least <math>k+1</math> points. Therefore, the same superlogarithmic dependence on <math>d</math> holds for the lower bound on the equilateral dimension of <math>d</math>-dimensional space.[8]
Riemannian manifolds
For any <math>d</math>-dimensional Riemannian manifold the equilateral dimension is at least <math>d+1</math>.[5] For a <math>d</math>-dimensional sphere, the equilateral dimension is <math>d+2</math>, 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
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