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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the equilateral dimension of a
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
is the maximum size of any subset of the space whose points are all at equal distances to each other. Equilateral dimension has also been called " metric dimension", but the term "metric dimension" also has many other inequivalent usages. The equilateral dimension of
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
is d+1, achieved by the vertices of a regular
simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
, and the equilateral dimension of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
with the
Chebyshev distance In mathematics, Chebyshev distance (or Tchebychev distance), maximum metric, or L∞ metric is a metric defined on a real coordinate space where the distance between two points is the greatest of their differences along any coordinate dimensio ...
(L^\infty norm) is 2^d, achieved by the vertices of a
hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...
. However, the equilateral dimension of a space with the
Manhattan distance Taxicab geometry or Manhattan geometry is geometry where the familiar Euclidean distance is ignored, and the distance between two point (geometry), points is instead defined to be the sum of the absolute differences of their respective Cartesian ...
(L^1 norm) is not known. Kusner's conjecture, named after Robert B. Kusner, states that it is exactly 2d, achieved by the vertices of a cross polytope.


Lebesgue spaces

The equilateral dimension has been particularly studied for Lebesgue spaces, finite-dimensional
normed vector space The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898. The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war ...
s with the L^p norm \ \, x\, _p=\bigl(, x_1, ^p+, x_2, ^p+\cdots+, x_d, ^p\bigr)^. The equilateral dimension of L^p spaces of dimension d behaves differently depending on the value of p: *For p=1, the L^p norm gives rise to
Manhattan distance Taxicab geometry or Manhattan geometry is geometry where the familiar Euclidean distance is ignored, and the distance between two point (geometry), points is instead defined to be the sum of the absolute differences of their respective Cartesian ...
. In this case, it is possible to find 2d equidistant points, the vertices of an axis-aligned cross polytope. The equilateral dimension is known to be exactly 2d for d\le 4, and to be upper bounded by O(d\log d) for all d.. Robert B. Kusner suggested in 1983 that the equilateral dimension for this case should be exactly 2d; this suggestion (together with a related suggestion for the equilateral dimension when p>2) has come to be known as Kusner's conjecture. *For 1, the equilateral dimension is at least (1+\varepsilon)d where \varepsilon is a constant that depends on p.. *For p=2, the L^p norm is the familiar
Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is o ...
. The equilateral dimension of d-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
is d+1: the d+1 vertices of an
equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
,
regular tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...
, or higher-dimensional regular
simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
form an equilateral set, and every equilateral set must have this form.. *For 2, the equilateral dimension is at least d+1: for instance the d
basis vector In mathematics, a set of elements of a vector space is called a basis (: bases) if every element of can be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as ...
s of the vector space together with another vector of the form (-x,-x,\dots) for a suitable choice of x form an equilateral set. Kusner's conjecture states that in these cases the equilateral dimension is exactly d+1. Kusner's conjecture has been proven for the special case that p=4. When p is an odd integer the equilateral dimension is upper bounded by O(d\log d). *For p=\infty (the limiting case of the L^p norm for finite values of p, in the limit as p grows to infinity) the L^p norm becomes the
Chebyshev distance In mathematics, Chebyshev distance (or Tchebychev distance), maximum metric, or L∞ metric is a metric defined on a real coordinate space where the distance between two points is the greatest of their differences along any coordinate dimensio ...
, the maximum absolute value of the differences of the coordinates. For a d-dimensional vector space with the Chebyshev distance, the equilateral dimension is 2^d: the 2^d vertices of an axis-aligned
hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...
are at equal distances from each other, and no larger equilateral set is possible.


Normed vector spaces

Equilateral dimension has also been considered for
normed vector space The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898. The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war ...
s with norms other than the L^p norms. The problem of determining the equilateral dimension for a given norm is closely related to the
kissing number problem In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sphere. For a given sphere packing (arrangement o ...
: 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 2^d; that is, the L^\infty norm has the highest equilateral dimension among all normed spaces.. 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 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 L^p norm, Petty's question has a positive answer: the equilateral dimension is at least d+1.. 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.; . more specifically, according to a variation of Dvoretzky's theorem by , 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) 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.


Riemannian manifolds

For any d-dimensional
Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
the equilateral dimension is at least d+1. For a d-dimensional
sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
, the equilateral dimension is d+2, the same as for a Euclidean space of one higher dimension into which the sphere can be embedded. 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.


Notes


References

*. *. *. *. *. *. *. *. *. *. {{refend Metric geometry Dimension theory