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Busemann–Petty Problem
In the mathematical field of convex geometry, the Busemann–Petty problem, introduced by , asks whether it is true that a symmetric convex body with larger central hyperplane sections has larger volume. More precisely, if ''K'', ''T'' are symmetric convex bodies in R''n'' such that : \mathrm_ \, (K \cap A) \leq \mathrm_ \, (T \cap A) for every hyperplane ''A'' passing through the origin, is it true that Vol''n'' ''K'' ≤ Vol''n'' ''T''? Busemann and Petty showed that the answer is positive if ''K'' is a ball. In general, the answer is positive in dimensions at most 4, and negative in dimensions at least 5. History showed that the Busemann–Petty problem has a negative solution in dimensions at least 12, and this bound was reduced to dimensions at least 5 by several other authors. pointed out a particularly simple counterexample: all sections of the unit volume cube have measure at most , while in dimensions at least 10 all central sections of the u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Convex Body
In mathematics, a convex body in n-dimensional Euclidean space \R^n is a compact convex set with non-empty interior. A convex body K is called symmetric if it is centrally symmetric with respect to the origin; that is to say, a point x lies in K if and only if its antipode, - x also lies in K. Symmetric convex bodies are in a one-to-one correspondence with the unit balls of norms on \R^n. Important examples of convex bodies are the Euclidean ball, the hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, ... and the cross-polytope. See also * * References * {{Authority control Multi-dimensional geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersection Body
In convex geometry, the projection body \Pi K of a convex body K in ''n''-dimensional Euclidean space is the convex body such that for any vector u\in S^, the support function of \Pi K in the direction ''u'' is the (''n'' – 1)-dimensional volume of the projection of ''K'' onto the hyperplane orthogonal to ''u''. Minkowski showed that the projection body of a convex body is convex. and used projection bodies in their solution to Shephard's problem. For K a convex body, let \Pi^\circ K denote the polar body of its projection body. There are two remarkable affine isoperimetric inequality for this body. proved that for all convex bodies K, : V_n(K)^ V_n(\Pi^\circ K)\le V_n(B^n)^ V_n(\Pi^\circ B^n), where B^n denotes the ''n''-dimensional unit ball and V_n is ''n''-dimensional volume, and there is equality precisely for ellipsoids. proved that for all convex bodies K, : V_n(K)^ V_n(\Pi^\circ K)\ge V_n(T^n)^ V_n(\Pi^\circ T^n), where T^n denotes any n-dimensional simplex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lp Norm
In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourbaki group they were first introduced by Frigyes Riesz . spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines. Applications Statistics In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, are defined in terms of metrics, and measures of central tendency can be characterized as solutions to variational problems. In penalized regression, "L1 penalty" and "L2 penalty" refer to penaliz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shephard's Problem
In mathematics, Shephard's problem, is the following geometrical question asked by Geoffrey Colin Shephard in 1964: if ''K'' and ''L'' are centrally symmetric convex bodies In mathematics, a convex body in n-dimensional Euclidean space \R^n is a compact convex set with non-empty interior. A convex body K is called symmetric if it is centrally symmetric with respect to the origin; that is to say, a point x lies in ... in ''n''-dimensional Euclidean space such that whenever ''K'' and ''L'' are projection (mathematics), projected onto a hyperplane, the volume of the projection of ''K'' is smaller than the volume of the projection of ''L'', then does it follow that the volume of ''K'' is smaller than that of ''L''? In this case, "centrally symmetric" means that the Reflection symmetry, reflection of ''K'' in the origin, ''−K'', is a translate of ''K'', and similarly for ''L''. If ''k'' : R''n'' → Π''k'' is a projection (mathematics), projection of R'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Journal Of Mathematics
The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United States, established in 1878 at the Johns Hopkins University by James Joseph Sylvester, an English-born mathematician who also served as the journal's editor-in-chief from its inception through early 1884. Initially W. E. Story was associate editor in charge; he was replaced by Thomas Craig in 1880. For volume 7 Simon Newcomb became chief editor with Craig managing until 1894. Then with volume 16 it was "Edited by Thomas Craig with the Co-operation of Simon Newcomb" until 1898. Other notable mathematicians who have served as editors or editorial associates of the journal include Frank Morley, Oscar Zariski, Lars Ahlfors, Hermann Weyl, Wei-Liang Chow, S. S. Chern, André Weil, Harish-Chandra, Jean Dieudonné, Henri Cartan, Stephen S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed at publishing articles addressed to a broader "mathematical community", and not only to mathematicians in the author's field. Herbert Busemann writes, in the preface of the first issue, "The need for expository articles addressing either all mathematicians or only those in somewhat related fields has long been felt, but little has been done outside of the USSR. The serial publication ''Advances in Mathematics'' was created in response to this demand." Abstracting and indexing The journal is abstracted and indexed in: * |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential in in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematika
''Mathematika'' is a peer-reviewed mathematics journal that publishes both pure and applied mathematical articles. The journal was founded by Harold Davenport in the 1950s. The journal is published by the London Mathematical Society, on behalf of the journal's owner University College London. Indexing and abstracting According to the ''Journal Citation Reports'', the journal has a 2020 impact factor of 0.844. The journal in indexing in the following bibliographic databases: * MathSciNet * Science Citation Index Expanded * Web of Science * Zentralblatt MATH zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastructur ... References {{reflist London Mathematical Society Mathematics education in the United Kingdom Mathematics journals Publications established in 1954 Quarterly journals W ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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