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Brunn–Minkowski Theorem
In mathematics, the Brunn–Minkowski theorem (or Brunn–Minkowski inequality) is an inequality relating the volumes (or more generally Lebesgue measures) of compact subsets of Euclidean space. The original version of the Brunn–Minkowski theorem (Hermann Brunn 1887; Hermann Minkowski 1896) applied to convex sets; the generalization to compact nonconvex sets stated here is due to Lazar Lyusternik (1935). Statement Let ''n'' ≥ 1 and let ''μ'' denote the Lebesgue measure on R''n''. Let ''A'' and ''B'' be two nonempty compact subsets of R''n''. Then the following inequality holds: : \mu (A + B) \geq mu (A) + mu (B), where ''A'' + ''B'' denotes the Minkowski sum: :A + B := \. The theorem is also true in the setting where A, B, A + B are only assumed to be measurable and non-empty. Multiplicative version The multiplicative form of Brunn–Minkowski inequality states that \mu(\lambda A + (1 - \lambda) B) \geq \mu(A)^ \mu(B)^ for all \lambda \in ,1. The Brunn–Minkowsk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Concentration Of Measure
In mathematics, concentration of measure (about a median) is a principle that is applied in measure theory, probability and combinatorics, and has consequences for other fields such as Banach space theory. Informally, it states that "A random variable that depends in a Lipschitz way on many independent variables (but not too much on any of them) is essentially constant". The concentration of measure phenomenon was put forth in the early 1970s by Vitali Milman in his works on the local theory of Banach spaces, extending an idea going back to the work of Paul Lévy. It was further developed in the works of Milman and Gromov, Maurey, Pisier, Schechtman, Talagrand, Ledoux, and others. The general setting Let (X, d) be a metric space with a measure \mu on the Borel sets with \mu(X) = 1. Let :\alpha(\varepsilon) = \sup \left\, where :A_\varepsilon = \left\ is the \varepsilon-''extension'' (also called \varepsilon-fattening in the context of the Hausdorff distance) of a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calculus Of Variations
The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of functionals: Map (mathematics), mappings from a set of Function (mathematics), functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as ''geodesics''. A related problem is posed by Fermat's principle: li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rolf Schneider
Rolf Georg Schneider (born 17 March 1940, Hagen, Germany) is a mathematician. Schneider is a professor emeritus at the University of Freiburg. His main research interests are convex geometry and stochastic geometry. Career Schneider completed his PhD 1967 with Ruth Moufang at Goethe University Frankfurt with a thesis titled (''Elliptisch gekrümmte Hyperflächen in der affinen Differentialgeometrie im Großen''). In 1969, he got his Habilitation in Bochum. In 1970, he was appointed as a full professor at Technische Universität Berlin and in 1974 at the University of Freiburg. He became a Fellow of the American Mathematical Society in 2014 and received an honorary doctorate of the University of Salzburg The University of Salzburg (, ), also known as the Paris Lodron University of Salzburg (''Paris-Lodron-Universität Salzburg'', PLUS), is an Austrian public university in Salzburg, Salzburg municipality, Salzburg (federal state), Salzburg State, ... in the same year. Research ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heinrich Guggenheimer
Heinrich Walter Guggenheimer (July 21, 1924 – March 4, 2021) was a Jewish, German-born Swiss-American mathematician who has contributed to knowledge in differential geometry, topology, algebraic geometry, and convexity. He has also contributed volumes on Jewish sacred literature. Guggenheimer was born in Nuremberg, Germany. He is the son of Marguerite Bloch and the physicist Dr. Siegfried Guggenheimer. He studied in Zürich, Switzerland at the , receiving his diploma in 1947 and a D.Sc. in 1951. His dissertation was titled "On complex analytic manifolds with Kahler metric". It was published in ''Commentarii Mathematici Helvetici'' (in German). Guggenheimer began his teaching career at the Hebrew University as a lecturer, 1954–56. He was a professor at the Bar Ilan University, 1956–59. In 1959, he immigrated to the United States, becoming a naturalized citizen in 1965. Washington State University was his first American post, where he was an associate professor. After one ye ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vitale's Random Brunn–Minkowski Inequality
In mathematics, Vitale's random Brunn–Minkowski inequality is a theorem due to Richard Vitale that generalizes the classical Brunn–Minkowski inequality for compact subsets of ''n''-dimensional Euclidean space R''n'' to random compact sets. Statement of the inequality Let ''X'' be a random compact set in R''n''; that is, a Borel–measurable function from some probability space (Ω, Σ, Pr) to the space of non-empty, compact subsets of R''n'' equipped with the Hausdorff metric. A random vector ''V'' : Ω → R''n'' is called a selection of ''X'' if Pr(''V'' ∈ ''X'') = 1. If ''K'' is a non-empty, compact subset of R''n'', let :\, K \, = \max \left\ and define the set-valued expectation E 'X''of ''X'' to be :\mathrm = \. Note that E 'X''is a subset of R''n''. In this notation, Vitale's random Brunn–Minkowski inequality is that, for any random compact set ''X'' with E X\, +\infty, :\left( \math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isoperimetric Inequality
In mathematics, the isoperimetric inequality is a geometric inequality involving the square of the circumference of a closed curve in the plane and the area of a plane region it encloses, as well as its various generalizations. '' Isoperimetric'' literally means "having the same perimeter". Specifically, the isoperimetric inequality states, for the length ''L'' of a closed curve and the area ''A'' of the planar region that it encloses, that :4\pi A \le L^2, and that equality holds if and only if the curve is a circle. The isoperimetric problem is to determine a plane figure of the largest possible area whose boundary has a specified length. The closely related ''Dido's problem'' asks for a region of the maximal area bounded by a straight line and a curvilinear arc whose endpoints belong to that line. It is named after Dido, the legendary founder and first queen of Carthage. The solution to the isoperimetric problem is given by a circle and was known already in Ancient Greece. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brunn–Minkowski Theorem
In mathematics, the Brunn–Minkowski theorem (or Brunn–Minkowski inequality) is an inequality relating the volumes (or more generally Lebesgue measures) of compact subsets of Euclidean space. The original version of the Brunn–Minkowski theorem (Hermann Brunn 1887; Hermann Minkowski 1896) applied to convex sets; the generalization to compact nonconvex sets stated here is due to Lazar Lyusternik (1935). Statement Let ''n'' ≥ 1 and let ''μ'' denote the Lebesgue measure on R''n''. Let ''A'' and ''B'' be two nonempty compact subsets of R''n''. Then the following inequality holds: : \mu (A + B) \geq mu (A) + mu (B), where ''A'' + ''B'' denotes the Minkowski sum: :A + B := \. The theorem is also true in the setting where A, B, A + B are only assumed to be measurable and non-empty. Multiplicative version The multiplicative form of Brunn–Minkowski inequality states that \mu(\lambda A + (1 - \lambda) B) \geq \mu(A)^ \mu(B)^ for all \lambda \in ,1. The Brunn–Minkowsk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scaling (geometry)
In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a '' scale factor'' that is the same in all directions ( isotropically). The result of uniform scaling is similar (in the geometric sense) to the original. A scale factor of 1 is normally allowed, so that congruent shapes are also classed as similar. Uniform scaling happens, for example, when enlarging or reducing a photograph, or when creating a scale model of a building, car, airplane, etc. More general is scaling with a separate scale factor for each axis direction. Non-uniform scaling (anisotropic scaling) is obtained when at least one of the scaling factors is different from the others; a special case is directional scaling or stretching (in one direction). Non-uniform scaling changes the shape of the object; e.g. a square may change into a rectangle, or into a parallelogram if the sides of the square are not parallel to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Translation (geometry)
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same Distance geometry, distance in a given direction (geometry), direction. A translation can also be interpreted as the addition of a constant vector space, vector to every point, or as shifting the Origin (mathematics), origin of the coordinate system. In a Euclidean space, any translation is an isometry. As a function If \mathbf is a fixed vector, known as the ''translation vector'', and \mathbf is the initial position of some object, then the translation function T_ will work as T_(\mathbf)=\mathbf+\mathbf. If T is a translation, then the image (mathematics), image of a subset A under the function (mathematics), function T is the translate of A by T . The translate of A by T_ is often written as A+\mathbf . Application in classical physics In classical physics, translational motion is movement that changes the Position (geometry), positio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homothetic Transformation
In mathematics, a homothety (or homothecy, or homogeneous dilation) is a Transformation (mathematics), transformation of an affine space determined by a point called its ''center'' and a nonzero number called its ''ratio'', which sends point to a point by the rule, : \overrightarrow=k\overrightarrow for a fixed number k\ne 0. Using position vectors: :\mathbf x'=\mathbf s + k(\mathbf x -\mathbf s). In case of S=O (Origin): :\mathbf x'=k\mathbf x, which is a uniform scaling and shows the meaning of special choices for k: :for k=1 one gets the ''identity'' mapping, :for k=-1 one gets the ''reflection'' at the center, For 1/k one gets the ''inverse'' mapping defined by k. In Euclidean geometry homotheties are the Similarity (geometry), similarities that fix a point and either preserve (if k>0) or reverse (if k<0) the direction of all vectors. Together with the Translation (geometry), translations, all homotheties of an affine (or Euclidean) space form a group (mathematics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
