Extrinsically, branched n-manifolds are n-dimensional complexes embedded into some Euclidean space such that each point has a well-defined n-dimensional tangent space.
A typical setting is a finite family of convex figures in the Euclidean space Rd that are homothetic to a given X, for example, a square as in the original question, a disk, or a d-dimensional cube.
David Preiss is a professor of mathematics at the University of Warwick and the winner of the 2008 LMS Pólya Prize for his 1987 result on Geometry of Measures, where he solved the remaining problem in the geometric theoretic structure of sets and measures in Euclidean space.
Intuitively, a flat manifold is one that "locally looks like" Euclidean space in terms of distances and angles, e.g. the interior angles of a triangle add up to 180°.
In mathematics, the Johnson–Lindenstrauss lemma is a result named after William B. Johnson and Joram Lindenstrauss concerning low-distortion embeddings of points from high-dimensional into low-dimensional Euclidean space.
If R denotes the set of all real numbers, then R2 := R × R represents the Euclidean plane and R3 := R × R × R represents three-dimensional Euclidean space.
Three-dimensional Euclidean space is irreducible: all smooth 2-spheres in it bound balls.
Since Ricci curvature measures the amount by which the volume of a small geodesic ball deviates from the volume of a ball in Euclidean space, small geodesic balls will have no volume deviation, but their "shape" may vary from the shape of the standard ball in Euclidean space.
Manifolds are often defined as embedded submanifolds of Euclidean space Rn, so this forms a very important special case.
In mathematics, the Vitali covering lemma is a combinatorial and geometric result commonly used in measure theory of Euclidean spaces.
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Finite version: Let be any finite collection of balls contained in d-dimensional Euclidean space Rd (or, more generally, in an arbitrary metric space).
In mathematical analysis, the Whitney covering lemma asserts the existence of a certain type of partition of an open set in a Euclidean space.
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The Euclidean shortest path problem is a problem in computational geometry: given a set of polyhedral obstacles in a Euclidean space, and two points, find the shortest path between the points that does not intersect any of the obstacles.
In mathematics, the Gaussian isoperimetric inequality, proved by Boris Tsirelson and Vladimir Sudakov and independently by Christer Borell, states that among all sets of given Gaussian measure in the n-dimensional Euclidean space, half-spaces have the minimal Gaussian boundary measure.
Examples of topologies include the Zariski topology in algebraic geometry that reflects the algebraic nature of varieties, and the topology on a differential manifold in differential topology where each point within the space is contained in an open set that is homeomorphic to an open ball in a finite-dimensional Euclidean space.