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In physics and mathematics, the dimension of a mathematical space is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. A two-dimensional Euclidean space is a two-dimensional space on the plane. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces.
In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry.
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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. Equilateral dimension has also been called "metric dimension", but the term "metric dimension" also has many other inequivalent usages. The equilateral dimension of a -dimensional Euclidean space is , achieved by a regular simplex, and the equilateral dimension of a -dimensional vector space with the Chebyshev distance is , achieved by a hypercube. However, the equilateral dimension of a space with the Manhattan distance is not known; Kusner's conjecture, named after Robert B. Kusner, states that it is exactly , achieved by a cross polytope.
In functional analysis, a branch of mathematics, the distortion problem is to determine by how much one can distort the unit sphere in a given Banach space using an equivalent norm. Specifically, a Banach space X is called λ-distortable if there exists an equivalent norm |x| on X such that, for all infinite-dimensional subspaces Y in X,
In geometry of normed spaces, the Holmes–Thompson volume is a notion of volume that allows to compare sets contained in different normed spaces. It was introduced by Raymond D. Holmes and Anthony Charles Thompson.
References
- 1 2 Schäffer, Juan Jorge (1967), "Inner diameter, perimeter, and girth of spheres", Mathematische Annalen , 173: 79–82, doi:10.1007/BF01351519, MR 0218875, S2CID 116325652 .
- 1 2 3 Álvarez Paiva, J. C. (2006), "Some problems on Finsler geometry", Handbook of differential geometry. Vol. II, Elsevier/North-Holland, Amsterdam, pp. 1–33, doi:10.1016/S1874-5741(06)80004-X, MR 2194667 . See in particular p. 16.
- ↑ Harrell, R. E.; Karlovitz, L. A. (1970), "Girths and flat Banach spaces", Bulletin of the American Mathematical Society, 76 (6): 1288–1291, doi: 10.1090/S0002-9904-1970-12643-X , MR 0267383 .
- ↑ Álvarez Paiva, J. C. (2006), "Dual spheres have the same girth" (PDF), American Journal of Mathematics, 128 (2): 361–371, arXiv: math/0408414 , doi:10.1353/ajm.2006.0015, MR 2214896, S2CID 201749975 .
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