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Helly's Theorem
Helly's theorem is a basic result in discrete geometry on the intersection of convex sets. It was discovered by Eduard Helly in 1913,. but not published by him until 1923, by which time alternative proofs by and had already appeared. Helly's theorem gave rise to the notion of a Helly family. Statement Let be a finite collection of convex subsets of , with . If the intersection of every of these sets is nonempty, then the whole collection has a nonempty intersection; that is, :\bigcap_^n X_j\ne\varnothing. For infinite collections one has to assume compactness: Let be a collection of compact convex subsets of , such that every subcollection of cardinality at most has nonempty intersection. Then the whole collection has nonempty intersection. Proof We prove the finite version, using Radon's theorem as in the proof by . The infinite version then follows by the finite intersection property characterization of compactness: a collection of closed subsets of a compact spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helly's Theorem
Helly's theorem is a basic result in discrete geometry on the intersection of convex sets. It was discovered by Eduard Helly in 1913,. but not published by him until 1923, by which time alternative proofs by and had already appeared. Helly's theorem gave rise to the notion of a Helly family. Statement Let be a finite collection of convex subsets of , with . If the intersection of every of these sets is nonempty, then the whole collection has a nonempty intersection; that is, :\bigcap_^n X_j\ne\varnothing. For infinite collections one has to assume compactness: Let be a collection of compact convex subsets of , such that every subcollection of cardinality at most has nonempty intersection. Then the whole collection has nonempty intersection. Proof We prove the finite version, using Radon's theorem as in the proof by . The infinite version then follows by the finite intersection property characterization of compactness: a collection of closed subsets of a compact spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carathéodory's Theorem (convex Hull)
Carathéodory's theorem is a theorem in convex geometry. It states that if a point x lies in the convex hull \mathrm(P) of a set P\subset \R^d, then x can be written as the convex combination of at most d+1 points in P. More sharply, x can be written as the convex combination of at most d+1 ''extremal'' points in P, as non-extremal points can be removed from P without changing the membership of ''x'' in the convex hull. Its equivalent theorem for conical combinations states that if a point x lies in the conical hull \mathrm(P) of a set P\subset \R^d, then x can be written as the conical combination of at most d points in P. The similar theorems of Helly and Radon are closely related to Carathéodory's theorem: the latter theorem can be used to prove the former theorems and vice versa. The result is named for Constantin Carathéodory, who proved the theorem in 1911 for the case when P is compact. In 1914 Ernst Steinitz expanded Carathéodory's theorem for arbitrary set. Exa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Discrete Geometry
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Convex Geometry
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematische Annalen
''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück, and Nigel Hitchin. Currently, the managing editor of Mathematische Annalen is Thomas Schick. Volumes 1–80 (1869–1919) were published by Teubner. Since 1920 (vol. 81), the journal has been published by Springer. In the late 1920s, under the editorship of Hilbert, the journal became embroiled in controversy over the participation of L. E. J. Brouwer on its editorial board, a spillover from the foundational Brouwer–Hilbert controversy. Between 1945 and 1947 the journal briefly ceased publication. References External links''Mathematische Annalen''homepage at Springer''Mathematische Annalen''archive (1869†... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heinrich Guggenheimer
Heinrich Walter Guggenheimer (July 21, 1924 – March 4, 2021) was a 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. Heinrich Guggenheimer was born in Nuremberg, Germany. He is the son of Marguerite Bloch and the Physicist Dr. Siegfried Guggenheimer. He studied in Zurich, Switzerland at the Eidgenössiche Technische Hochschule, 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:25:257–97 (in German). Guggenheimer began his teaching career at Hebrew University as lecturer 1954–56. He was a professor at 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Tverberg's Theorem
In discrete geometry, Tverberg's theorem, first stated by , is the result that sufficiently many points in ''d''-dimensional Euclidean space can be partitioned into subsets with intersecting convex hulls. Specifically, for any set of :(d + 1)(r - 1) + 1\ points there exists a point ''x'' (not necessarily one of the given points) and a partition of the given points into ''r'' subsets, such that ''x'' belongs to the convex hull of all of the subsets. The partition resulting from this theorem is known as a Tverberg partition. Examples For ''r'' = 2, Tverberg's theorem states that any ''d'' + 2 points may be partitioned into two subsets with intersecting convex hulls; this special case is known as Radon's theorem. In this case, for points in general position, there is a unique partition. The case ''r'' = 3 and ''d'' = 2 states that any seven points in the plane may be partitioned into three subsets with intersecting convex hulls. The illustrat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Choquet Theory
In mathematics, Choquet theory, named after Gustave Choquet, is an area of functional analysis and convex analysis concerned with measures which have support on the extreme points of a convex set ''C''. Roughly speaking, every vector of ''C'' should appear as a weighted average of extreme points, a concept made more precise by generalizing the notion of weighted average from a convex combination to an integral taken over the set ''E'' of extreme points. Here ''C'' is a subset of a real vector space ''V'', and the main thrust of the theory is to treat the cases where ''V'' is an infinite-dimensional (locally convex Hausdorff) topological vector space along lines similar to the finite-dimensional case. The main concerns of Gustave Choquet were in potential theory. Choquet theory has become a general paradigm, particularly for treating convex cones as determined by their extreme rays, and so for many different notions of ''positivity'' in mathematics. The two ends of a line segment dete ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Krein–Milman Theorem
In the mathematical theory of functional analysis, the Krein–Milman theorem is a proposition about compact convex sets in locally convex topological vector spaces (TVSs). This theorem generalizes to infinite-dimensional spaces and to arbitrary compact convex sets the following basic observation: a convex (i.e. "filled") triangle, including its perimeter and the area "inside of it", is equal to the convex hull of its three vertices, where these vertices are exactly the extreme points of this shape. This observation also holds for any other convex polygon in the plane \R^2. Statement and definitions Preliminaries and definitions Throughout, X will be a real or complex vector space. For any elements x and y in a vector space, the set , y:= \ is called the or closed interval between x and y. The or open interval between x and y is (x, x) := \varnothing when x = y while it is (x, y) := \ when x \neq y; it satisfies (x, y) = , y\setminus \ and , y= (x, y) \cup \. The points x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shapley–Folkman Lemma
The Shapley–Folkman lemma is a result in convex geometry that describes the Minkowski addition of sets in a vector space. It is named after mathematicians Lloyd Shapley and Jon Folkman, but was first published by the economist Ross M. Starr. The lemma may be intuitively understood as saying that, if the number of summed sets exceeds the dimension of the vector space, then their Minkowski sum is approximately convex. Related results provide more refined statements about ''how close'' the approximation is. For example, the Shapley–Folkman theorem provides an upper bound on the distance between any point in the Minkowski sum and its convex hull. This upper bound is sharpened by the Shapley–Folkman–Starr theorem (alternatively, Starr's corollary). The Shapley–Folkman lemma has applications in economics, optimization and probability theory. In economics, it can be used to extend results proved for convex preferences to non-convex preferences. In op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kirchberger's Theorem
Kirchberger's theorem is a theorem in discrete geometry, on linear separability. The two-dimensional version of the theorem states that, if a finite set of red and blue points in the Euclidean plane has the property that, for every four points, there exists a line separating the red and blue points within those four, then there exists a single line separating all the red points from all the blue points. Donald Watson phrases this result more colorfully, with a farmyard analogy: More generally, for finitely many red and blue points in d-dimensional Euclidean space, if the red and blue points in every subset of d+2 of the points are linearly separable, then all the red points and all the blue points are linearly separable. Another equivalent way of stating the result is that, if the convex hulls of finitely many red and blue points have a nonempty intersection, then there exists a subset of d+2 points for which the convex hulls of the red and blue points in the subsets also intersect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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