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Convex Geometry
In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming, probability theory, game theory, etc. Classification According to the Mathematics Subject Classification MSC2010, the mathematical discipline ''Convex and Discrete Geometry'' includes three major branches: * general convexity * polytopes and polyhedra * discrete geometry (though only portions of the latter two are included in convex geometry). General convexity is further subdivided as follows: *axiomatic and generalized convexity *convex sets without dimension restrictions *convex sets in topological vector spaces *convex sets in 2 dimensions (including convex curves) *convex sets in 3 dimensions (including convex surfaces) *convex sets in ''n'' dimensions (including convex hy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mixed Volume
In mathematics, more specifically, in convex geometry, the mixed volume is a way to associate a non-negative number to an of convex bodies in space. This number depends on the size and shape of the bodies and on their relative orientation to each other. Definition Let K_1, K_2, \dots, K_r be convex bodies in \mathbb^n and consider the function : f(\lambda_1, \ldots, \lambda_r) = \mathrm_n (\lambda_1 K_1 + \cdots + \lambda_r K_r), \qquad \lambda_i \geq 0, where \text_n stands for the n-dimensional volume and its argument is the Minkowski sum of the scaled convex bodies K_i. One can show that f is a homogeneous polynomial of degree n, therefore it can be written as : f(\lambda_1, \ldots, \lambda_r) = \sum_^r V(K_, \ldots, K_) \lambda_ \cdots \lambda_, where the functions V are symmetric. For a particular index function j \in \^n , the coefficient V(K_, \dots, K_) is called the mixed volume of K_, \dots, K_. Properties * The mixed volume is uniquely determined by the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Convexity Topics
This is a list of convexity topics, by Wikipedia page. * Alpha blending - the process of combining a translucent foreground color with a background color, thereby producing a new blended color. This is a convex combination of two colors allowing for transparency effects in computer graphics. * Barycentric coordinates - a coordinate system in which the location of a point of a simplex (a triangle, tetrahedron, etc.) is specified as the center of mass, or barycenter, of masses placed at its vertices. The coordinates are non-negative for points in the convex hull. * Borsuk's conjecture - a conjecture about the number of pieces required to cover a body with a larger diameter. Solved by Hadwiger for the case of smooth convex bodies. * Bond convexity - a measure of the non-linear relationship between price and yield duration of a bond to changes in interest rates, the second derivative of the price of the bond with respect to interest rates. A basic form of convexity in finance. * Carathé ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Werner Fenchel
Moritz Werner Fenchel (; 3 May 1905 – 24 January 1988) was a mathematician known for his contributions to geometry and to optimization theory. Fenchel established the basic results of convex analysis and nonlinear optimization theory which would, in time, serve as the foundation for nonlinear programming. A German-born Jew and early refugee from Nazi suppression of intellectuals, Fenchel lived most of his life in Denmark. Fenchel's monographs and lecture notes are considered influential. Biography Early life and education Fenchel was born on 3 May 1905 in Berlin, Germany, his younger brother was the Israeli film director and architect Heinz Fenchel. Fenchel studied mathematics and physics at the University of Berlin between 1923 and 1928. He wrote his doctorate thesis in geometry (''Über Krümmung und Windung geschlossener Raumkurven'') under Ludwig Bieberbach. Professorship in Germany From 1928 to 1933, Fenchel was Professor E. Landau's Assistant at the Univ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tommy Bonnesen
Tommy Bonnesen (27 March 1873 – 14 March 1935) was a Danish mathematician, known for Bonnesen's inequality. Bonnesen studied at the University of Copenhagen, where in 1902 he received his Ph.D. (promotion) with thesis ''Analytiske studier over ikke-euklidisk geometri'' (Analytic studies of non-Euclidean geometry). He was the Professor for Descriptive Geometry at the Polytekniske Læreanstalt. He did research on convex geometry and wrote a book on this subject with his student Werner Fenchel. Bonessen was an Invited Speaker at the ICM in 1924 in Toronto and in 1928 in Bologna. With Harald Bohr he was for many years the co-editor-in-chief of the Matematisk Tidsskrift of the Danish Mathematical Society. His younger daughter was the theatrical and cinematic star Beatrice Bonnesen (1906–1979). His elder daughter Merete Bonnesen (1901–1980) was a journalist employed by the newspaper Politiken ''Politiken'' is a leading Danish daily broadsheet newspaper, published by J ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermann Minkowski
Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician and professor at Königsberg, Zürich and Göttingen. He created and developed the geometry of numbers and used geometrical methods to solve problems in number theory, mathematical physics, and the theory of relativity. Minkowski is perhaps best known for his foundational work describing space and time as a four-dimensional space, now known as "Minkowski spacetime", which facilitated geometric interpretations of Albert Einstein's special theory of relativity (1905). Personal life and family Hermann Minkowski was born in the town of Aleksota, the Suwałki Governorate, the Kingdom of Poland, part of the Russian Empire, to Lewin Boruch Minkowski, a merchant who subsidized the building of the choral synagogue in Kovno, and Rachel Taubmann, both of Jewish descent. Hermann was a younger brother of the medical researcher Oskar (born 1858). In different sources Minkowski's nationality is variously giv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermann Brunn
Karl Hermann Brunn (1 August 1862 – 20 September 1939) was a German mathematician, known for his work in convex geometry (see Brunn–Minkowski inequality) and in knot theory. Brunnian links are named after him, as his 1892 article "Über Verkettung" included examples of such links. Life and work Hermann Brunn was born in Rome, and grew up in Munich. He studied mathematics and physics at the Ludwig Maximilian University of Munich The Ludwig Maximilian University of Munich (simply University of Munich or LMU; german: Ludwig-Maximilians-Universität München) is a public research university in Munich, Germany. It is Germany's sixth-oldest university in continuous operatio ..., graduating in 1887 with the thesis ''Über Ovale und Eiflächen'' (About ovals and eggforms). He habilitated in 1889. References {{DEFAULTSORT:Brunn, Hermann Geometers 19th-century German mathematicians 20th-century German mathematicians 1939 deaths 1862 births Italian emigrants to Germany ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Considered the greatest mathematician of ancient history, and one of the greatest of all time,* * * * * * * * * * Archimedes anticipated modern calculus and analysis by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove a range of geometrical theorems. These include the area of a circle, the surface area and volume of a sphere, the area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral. Heath, Thomas L. 1897. ''Works of Archimedes''. Archimedes' other mathematical achievements include deriving an approximation of pi, defining and in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid
Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' treatise, which established the foundations of geometry that largely dominated the field until the early 19th century. His system, now referred to as Euclidean geometry, involved new innovations in combination with a synthesis of theories from earlier Greek mathematicians, including Eudoxus of Cnidus, Hippocrates of Chios, Thales and Theaetetus (mathematician), Theaetetus. With Archimedes and Apollonius of Perga, Euclid is generally considered among the greatest mathematicians of antiquity, and one of the most influential in the history of mathematics. Very little is known of Euclid's life, and most information comes from the philosophers Proclus and Pappus of Alexandria many centuries later. Until the early Renaissance he was often mistaken f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antimatroid
In mathematics, an antimatroid is a formal system that describes processes in which a set is built up by including elements one at a time, and in which an element, once available for inclusion, remains available until it is included. Antimatroids are commonly axiomatized in two equivalent ways, either as a set system modeling the possible states of such a process, or as a formal language modeling the different sequences in which elements may be included. Dilworth (1940) was the first to study antimatroids, using yet another axiomatization based on lattice theory, and they have been frequently rediscovered in other contexts. The axioms defining antimatroids as set systems are very similar to those of matroids, but whereas matroids are defined by an '' exchange axiom'', antimatroids are defined instead by an ''anti-exchange axiom'', from which their name derives. Antimatroids can be viewed as a special case of greedoids and of semimodular lattices, and as a generalization of partia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Valuation (geometry)
In geometry, a valuation is a finitely additive function on a collection of admissible subsets of a fixed set X with values in an abelian semigroup. For example, the Lebesgue measure is a valuation on finite unions of convex bodies (that is, non-empty compact convex sets) of Euclidean space \R^n. Other examples of valuations on finite unions of convex bodies are the surface area, the mean width, and the Euler characteristic. In the geometric setting, often continuity (or smoothness) conditions are imposed on valuations, but there are also purely discrete facets of the theory. In fact, the concept of valuation has its origin in the dissection theory of polytopes and in particular Hilbert's third problem, which has grown into a rich theory, heavily reliant on advanced tools from abstract algebra. Definition Let X be a set and \mathcal S be a collection of admissible subsets of X. A function \phi on \mathcal S with values in an abelian semigroup R is called a valuation if it satisfie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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