Convexgeometry is the branch of geometry studying convex set s, mainly in Euclidean space . Convex sets occur naturally in many areas of mathematics computational geometry , convex analysis , discrete geometry , functional analysis , geometry of numbers , integral geometry , linear programming , probability ... branches of the mathematical discipline Convex and Discrete Geometry are General Convexity , Polytopes and Polyhedra , Discrete Geometry. Further classification of General Convexity results in the following list 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 hypersurfaces finite dimensional Banach spaces random convex sets and integral geometry asymptotic theory of convex ... programs spherical and hyperbolic convexity The phrase convexgeometry is also used in combinatorics ... to convexgeometry date back to antiquity and can be traced in the works of Euclid and Archimedes , it became ... Fenchel W. Fenchel gave a comprehensive survey of convexgeometry in Euclidean space R sup n sup . Further development of convexgeometry in the 20th century and its relations to numerous mathematical disciplines are summarized in the Handbook of convexgeometry edited by P. M. Gruber and J. M. Wills. See also List of convexity topics References Expository articles on convexgeometry K. Ball, An elementary introduction to modern convexgeometry, in Flavors of Geometry, pp. 1 58, Math. Sci ... of convexgeometry. Vol. A. B, North Holland, Amsterdam, 1993. R. Schneider, Convex bodies the Brunn ... on history of convexgeometry W. Fenchel, Convexity through the ages, Danish Danish Mathematical ... editors , Handbook of convexgeometry. Vol. A, pp. 1 15, North Holland, Amsterdam, 1993. Category Convexgeometry ar de Konvexgeometrie es Geometr a convexa nl Convexe meetkunde ... more details
Image Convex polygon illustration1.png right thumb A convex set. wiktionary convex The word convex means curving out or bulging outward , as opposed to Concave disambiguation concave . Convex or convexity may refer to Mathematics Convex set , a set of points containing all line segments between each pair of its points Convex function , a function with the Epigraph mathematics epigraph forming a convex set Convex polytope , a polytope which forms a convex set. These include convex polygon s. Convex hull , the minimal convex set containing a set of points X Convex combination , a linear combination of points with non negative coefficients that sum up to 1 Convex conjugate , a generalization of the Legendre transformation Convex bipartite graph , a special kind of bipartite graph Convex polygon , a polygon which is not concave Convex plane graph , a plane graph with convex faces Convex optimization Economics Convexity in economics Convex preferences , a preference relation with convex upper contour sets Non convexity economics Finance Bond convexity , a measure of the sensitivity of the price of a bond to changes in interest rates Convexity finance , second derivatives in financial modeling generally Optics Convex lens , a lens with surfaces that curve outward Art Convex and Concave , a lithograph print by the artist M. C. Escher Proper names Convex Computer , a company that produced a number of vector supercomputers, bought by HP in 1995 Convex Software Library , a client side open source solution for Internet Explorer which uses a hidden Java applet to process XForms Convex application, an iPhone Ipad software created by Ergonotics to perform conversion. See also List of convexity topics Obtuse angle Kachurovskii s theorem disambig cs Konvexn de Konvex it Convesso lb Konvex ja no Konveks nn Konveks ro Convexitate fi Konveksisuus sv Konvex ... more details
MergeTo Convex set date December 2011 In mathematics , a convex curve is the boundary topology boundary of a convex set . See also Secant line Category Convexgeometry fr Courbe convexe ... more details
. Projective geometry Projective , Convexgeometryconvex and discrete geometry are three sub disciplines ... geometry has become closely connected with computational geometry , computer graphics , convexgeometry ... theorem , an important result in Euclidean geometry Euclidean and projective geometry . Image Oxyrhynchus ... fragment of Euclid s Elements Geometry lang grc wikt geo earth , wikt metria measurement ..., and the properties of space. Geometry arose independently in a number of early cultures as a body ... science emerging in the West as early as Thales 6th Century BC . By the 3rd century BC geometry was put into an axiomatic system axiomatic form by Euclid , whose treatment Euclidean geometry ... geometry in digital imaging . Academic Press . p.1. ISBN 0127039708 ref Archimedes developed ... works in the field of geometry is called a geometer. The introduction of coordinates by Ren Descartes and the concurrent development of algebra marked a new stage for geometry, since geometric figures, such as plane curve s, could now be represented analytic geometry analytically , i.e., with functions ... century. Furthermore, the theory of perspective graphical perspective showed that there is more to geometry than just the metric properties of figures perspective is the origin of projective geometry . The subject of geometry was further enriched by the study of intrinsic structure of geometric objects ... geometry . In Euclid s time there was no clear distinction between physical space and geometrical space. Since the 19th century discovery of non Euclidean geometry , the concept of space ... geometry considers manifold s, spaces that are considerably more abstract than the familiar ... with additional structure, allowing one to speak about length. Modern geometry has multiple strong bonds with physics , exemplified by the ties between pseudo Riemannian geometry and general relativity ... the visual nature of geometry makes it initially more accessible than other parts of mathematics ... more details
hull Carath odory s theorem convex hull Convex hull Simplex DEFAULTSORT Convex Combination Category Convexgeometry Category Mathematical analysis Category Convex hulls de Linearkombination Spezialf lle ...Unreferenced date December 2009 Image Convex combination illustration.svg right thumb Given three points math x 1, x 2, x 3 math in a plane as shown in the figure, the point math P math is a convex combination of the three points, while math Q math is not. br math Q math is however an affine combination of the three points, as their affine hull is the entire plane. In convexgeometry , a convex combination is a linear combination of point geometry points which can be vector geometric vector s, scalar mathematics scalars , or more generally points in an affine space where all coefficients are non ... , math in a real vector space , a convex combination of these points is a point of the form math alpha ... i ge 0 math and math alpha 1 alpha 2 cdots alpha n 1. math As a particular example, every convex combination of two points lies on the line segment between the points. All convex combinations are within the convex hull of the given points. In fact, the collection of all such convex combinations of points in the set constitutes the convex hull of the set. There exist subsets of a vector space that are not closed under linear combinations but are closed under convex combinations. For example, the interval math 0,1 math is convex but generates the real number line under linear combinations. Another example is the convex set of probability distribution s, as linear combinations preserve neither nonnegativity nor affinity i.e., having total integral one . Other objects Similarly, a convex combination ... combination Affine, conical, and convex combinations A conical combination is a linear combination with nonnegative coefficients Weighted mean s are functionally the same as convex combinations, but they use ... s are like convex combinations, but the coefficients are not required to be non negative. Hence ... more details
math such that math p notin math Conv P p is called a Vertex geometry vertex of Conv P . In fact, a convex ... geometry line , the convex hull is the line segment joining the outermost two points. In the planar ... of convex hulls Main Convex hull algorithms In computational geometry, a number of algorithms are known ... operators Category Convex hulls Category Convex analysis Category Computational geometry ar ...Technical date June 2011 Image Extreme points illustration.png thumb right The convex hull of the red set is the blue convex set . See also Convex set Convex combination In mathematics , the convex hull or convex envelope for a Set mathematics set of points X in a real number real vector space V for example, usual 2 or 3 dimensional space is the minimal convex set containing X . When the set X is a finite set finite subset of the Euclidean geometry plane , we may imagine stretching a rubber band ... taut, it encloses the convex hull of X . ref name NeringTucker Pages 195 196 Evar D. Nering and Albert ... 7 mr 1226891 ref harv ref The convex hull also has a real vector space linear algebraic characterization the convex hull of X is the set of all convex combination s of points in X . In computational geometry , a basic problem is finding the convex hull for a given finite set of points in the plane mathematics plane . Existence of the convex hull To show that the convex hull of a set X in a real vector space V exists, notice that X is contained in at least one convex set the whole space V , for example , and any intersection of convex sets containing X is also a convex set containing X . It is then clear that the convex hull is the intersection of all convex sets containing X . This can be used as an alternative definition of the convex hull. The convex hull operator Conv has ... S Conv T , and idempotence idempotent Conv Conv S Conv S . Thus, the convex hull operator is a proper hull operator. Algebraic characterization Algebraically, the convex hull ... more details
texts in discrete geometry , convex polytopes are often simply called polytopes . Gr nbaum points out that this is solely to avoid the endless repetition of the word convex , and that the discussion should throughout be understood as applying only to the convex variety. A polytope is called full dimensional ... convex polytope, the minimal H description is in fact unique and is given by the set of the facet geometry ... over other fields . The face lattice A Face geometry face of a convex polytope is any intersection ... Polytopes Category Convexgeometry ...Image 3dpoly.svg thumb right A 3 dimensional convex polytope A convex polytope is a special case of a polytope , having the additional property that it is also a convex set of points in the n dimensional space R sup n sup . ref name grun Some authors use the terms convex polytope and convex polyhedron ...?id ofrBsl61lq8C&pg PA67&dq 22unbounded convex polyhedron 22&sig ACfU3U1Yv3iG XIn3hiuh84nK2e8UIcdAA ... convex polytope will be used below whenever the boundedness is critical to the discussed issue. Yet other texts treat a convex n polytope as a surface or n 1 manifold. Convex polytopes play an important ... . A comprehensive and influential book in the subject, called Convex Polytopes , was published ... Many examples of bounded convex polytopes can be found in the article polyhedron . In the 2 dimensional ... shape the intersection of two non parallel half planes , a shape defined by a convex polygonal chain with two ray geometry ray s attached to its ends, and a convex polygon . Special cases of an unbounded convex polytope are a slab between two parallel hyperplanes, a wedge defined by two non parallel half space s, a polyhedral cylinder infinite prism geometry prism , and a polyhedral cone infinite prism geometry prism , defined by three or more half spaces passing through a common point. Definitions A convex polytope may be defined in a number of ways, depending on what is more suitable for the problem ... more details
In mathematics , a convex body in n dimension al Euclidean space R sup n sup is a compact space compact convex set with non empty set empty interior topology interior . A convex body K is called symmetric if it is centrally symmetric with respect to the origin, i.e. a point x lies in K if and only if its antipode , &minus x , also lies in K . Symmetric convex bodies are in a bijection one to one correspondence with the unit ball s of norm mathematics norms on R sup n sup . Important examples of convex bodies are the Euclidean ball , the hypercube and the cross polytope . References cite journal last Gardner first Richard J. title The Brunn Minkowski inequality journal Bulletin of the American Mathematical Society Bull. Amer. Math. Soc. N.S. volume 39 issue 3 year 2002 pages 355&ndash 405 electronic doi 10.1090 S0273 0979 02 00941 2 Category Multi dimensional geometry es Cuerpo convexo ... more details
In mathematics , a convex graph may be a convex bipartite graph a convex plane graph the graph of a function graph of a convex function disambig ... more details
, a solid cube geometry cube is convex, but anything that is hollow or has a dent in it, for example, a crescent shape, is not convex. The notion can be generalized to other spaces as described below. In vector spaces Image Convex supergraph.png right thumb A convex function function is convex ... sets is orthoconvex. Some other properties of convex sets are valid as well. Not Euclidean geometry ... , March 2010. DEFAULTSORT Convex Set Category Convexgeometry Category Mathematical analysis ...Image Convex polygon illustration1.png right thumb alt Illustration of a convex set, which looks somewhat like a disk A green convex set contains the black line segment joining the points x and y. The entire line segment lies in the interior of the convex set A convex set. Image Convex polygon illustration2.png right thumb alt Illustration of a non convex set, which looks somewhat like a boomerang or wedge. A green non convexconvex set contains the black line segment joining the points x and y. Part of the line segment lies outside of the green non convex set. A non convex set, with a line segment outside the set. In Euclidean space , an object is convex if for every pair of points within the object ... graph in blue , is a convex set. Let S be a vector space over the real number s, or, more generally, some ordered field . This includes Euclidean spaces. A set mathematics set C in S is said to be convex ... that a convex set in a real number real or complex number complex topological vector space is path connected , thus connected space connected . A set C is called absolutely convex if it is convex and balanced set balanced . The convex subset s of R the set of real numbers are simply the intervals of R . Some examples of convex subsets of the Euclidean space Euclidean plane are solid regular polygon s, solid triangles, and intersections of solid triangles. Some examples of convex subsets ... s. The Kepler Poinsot polyhedra are examples of non convex sets. Properties If math S math is a convex ... more details
Unreferenced date December 2009 In linear algebra , a convex cone is a subset of a vector space over .... Image Convex cone illust.svg right thumb A convex cone light blue . Inside of it, the light red convex cone consists of all points x y with > 0 and > 0, for the depicted x and y . The curves ... of a vector space V is a convex cone if x y belongs to C , for any positive scalars , , and any ... subspace null vector vector space 0 are convex cones by this definition. Other examples are the set ... vectors x such that is a positive scalar and x is an element of some convex set convex subset X ... not contain 0 , this construction gives an open resp. closed convex circular cone . The intersection of two convex cones in the same vector space is again a convex cone, but their union may fail to be one. The class of convex cones is also closed under arbitrary linear map s. In particular, if C is a convex cone, so is its opposite C and C math cap math C is the largest linear subspace contained in C . Convex cones are linear cones If C is a convex cone, then for any positive scalar and any x in C the vector x 2 x 2 x is in C . It follows that a convex cone C is a special case of a cone linear algebra linear cone . Alternative definitions It follows from the above property that a convex cone can also be defined as a linear cone that is closed under convex combination s, or just under addition s. More succinctly, a set C is a convex cone if and only if C C and C C C , for any ... of convex cone by non negative scalars , , not both zero . Blunt and pointed cones According to the above definition, if C is a convex cone, then C math cup math 0 is a convex cone, too. A convex cone is said to be pointed or blunt depending on whether it includes the null vector 0 or not. Blunt cones can be excluded from the definition of convex cone by substituting non negative ... are convex cones. Moreover, any convex cone C that is not the whole space V must be contained in some ... more details
wiktionarypar plano convex Plano convex may refer to Plano convex lenses, in optics see Lens optics Types of simple lenses The plano convex type of mudbrick , used by the ancient Sumerians disambig ... more details
Image ConvexFunction.svg thumb 300px right Convex function on an interval. Image Epigraph convex.svg right thumb 300px A function in black is convex if and only if the region above its Graph of a function graph in green is a convex set . mergefrom Proper convex function discuss Talk Convex function Proper ... on an interval mathematics interval is called convex or convex downward or concave upward if the graph ..., a function is convex if its epigraph mathematics epigraph the set of points on or above the graph of the function is a convex set . More generally, this definition of convex functions makes sense for functions defined on a convex subset of any vector space . Convex functions play an important role ... where they are distinguished by a number of convenient properties. For instance, a strictly convex function ... additional hypotheses, convex functions continue to satisfy such properties and, as a result, they are the most well understood functionals in the calculus of variations . In probability theory , a convex ... value of the convex function of the random variable. This result, known as Jensen s inequality ... defined on a convex set X in a vector space is called convex if, for any two points math x 1 math and math ... is called strictly convex if math f tx 1 1 t x 2 t f x 1 1 t f x 2 , math for every math ... function concave if &minus f is strictly convex. Properties Suppose f is a function of one real ... R is symmetric in math x 1,x 2 math . f is convex if and only if math R x 1,x 2 math is monotonically ... of convexity is quite useful to prove the following results. A convex function f defined on some ... in the examples section . A function is midpoint convex on an interval C if math f left frac x 1 x ... that is midpoint convex will be convex. ref Sierpinski Theorem, Donoghue 1969 , http books.google.com ... convex will be convex. A differentiable function of one variable is convex on an interval if and only ... and convex then it is also continuously differentiable . A continuously differentiable function ... more details
In economics , convex preferences refer to a property of an individual s ordering of various outcomes which roughly corresponds to the idea that averages are better than the extremes . The concept roughly corresponds to the concept of marginal utility Diminishing marginal utility diminishing marginal utility but uses modern theory to represent the concept without requiring the use of utility function s. Comparable to the greater than or equal to Order theory Partially ordered sets ordering relation math geq math for real numbers, the notation math succeq math below can be translated as is at least as good as in Preference economics preference satisfaction . Use x , y , and z to denote three consumption bundles combinations of various quantities of various goods . Formally, a preference relation P on the consumption set X is Convex set convex if for any math x, y, z in X math where math y succeq x math and math z succeq x math , it is the case that math theta y 1 theta z succeq x math for any math theta in 0,1 math . That is, the preference ordering P is convex if for any two goods bundles that are each viewed as being at least as good as a third bundle, a weighted average of the two bundles is also viewed as being at least as good as the third bundle. Moreover, math P math is strictly convex if for any math x, y, z in X math where math y succeq x math , math z succeq x math , and math ... ordering P is strictly convex if for any two distinct goods bundles that are each viewed ... amount of each bundle is viewed as being better than the third bundle. A set of Convex function convex shaped indifference curve s displays convex preferences Given a convex indifference curve ... is a convex set . Convex preferences with their associated convex indifference mapping arise from Quasi convex function quasi concave utility functions, although these are not necessary for the analysis ... 9 See also Convex function Level set Quasi convex function Semi continuous function Shapley Folkman ... more details
Convex analysis is the branch of mathematics devoted to the study of properties of convex function s and convex set s, often with applications in convex optimization convex minimization , a subdomain of optimization mathematics optimization theory . Convex Sets main Convex set A convex set is a set math ... Rockafellar, R. Tyrrell title Convex Analysis publisher Princeton University Press location Princeton, NJ year 1997 origyear 1970 isbn 9780691015866 ref Convex Functions main Convex function A convex ..., a convex function is any extended real valued function such that its epigraph mathematics epigraph math left x,r in X times mathbb R f x leq r right math is a convex set. ref name Rockafellar Convex Conjugate main Convex conjugate The convex conjugate of an extended real valued not necessarily convex function math f X to mathbb R cap pm infty math is math f X to mathbb R cap pm infty math ... from the Fenchel Young inequality . For Proper convex function proper functions , math f f math if and only if math f math is convex and lower semi continuous . ref name Zalinescu rp pp.75 79 Convex Minimization main Convex optimization A convex minimization primal problem is one of the form math inf x in M f x math such that math f X to mathbb R cap pm infty math is a convex function and math M subseteq X math is a convex set. Dual Problem main Duality optimization Copied from Duality optimization ... given two dual pair s separated space separated locally convex space s math left X,X right math ... constraints math where math I math is the Characteristic function convex analysis indicator function ... where math F math is the convex conjugate in both variables of math F math . The duality gap is the difference ... point regularity conditions in convex optimization. Applications of the duality theory to enlargements ... Convex analysis in general vector spaces publisher World Scientific Publishing Co., Inc ... F math is the convex conjugate biconjugate of math F math Citation needed date January 2012 the primal ... more details
More footnotes date February 2012 Convex minimization , a subfield of mathematical optimization optimization , studies the problem of minimizing convex function s over convex set s. The convexity property ... math together with a convex function convex , real valued function mathematics function math f mathcal X to mathbb R math defined on a convex set convex subset math mathcal X math of math X math , the problem ... X math . The convexity of math f math makes the powerful tools of convex analysis applicable. In locally convex F finite dimensional normed space s, the Hahn Banach theorem and the existence ... computational methods. Convex minimization has applications in a wide range of disciplines, such as automatic ... improvements in computing and in optimization theory, convex minimization is nearly as straightforward as linear programming . Many optimization problems can be reformulated as convex minimization problems ... as a problem of minimizing the function f , which is convex . Convex optimization problem An optimization ... math is the objective , is called convex if math mathcal X math is a closed convex set and math f x math is convex on math mathbb R n math . ref cite book last1 Hiriart Urruty first1 Jean Baptiste last2 Lemar chal first2 Claude title Convex analysis and minimization algorithms Fundamentals page 291 year 1996 url http books.google.de books?id Gdl4Jc3RVjcC&printsec frontcover&dq lemarechal convex analysis and minimization&hl de&sa X&ei E602T4GXGMzQsgaPtJ2VDA&ved 0CDUQ6AEwAA v onepage&q convex 20minimization ... title Lectures on modern convex optimization analysis, algorithms, and engineering applications ... on Modern Convex Optimization Analysis, Algorithms,&hl de&sa X&ei 26c2T6G7HYrIswac0d2uDA&ved 0CDIQ6AEwAA v onepage&q convex 20programming&f false ref Alternatively, an optimization problem on the form ... i 1, dots,m end align math is called convex if the functions math f, g 1 ldots g m mathbb R n rightarrow mathbb R math are convex. ref Boyd Vandenberghe, p. 7 ref Theory The following statements ... more details
Original research date November 2010 Convex Computer Corporation was a company that developed, manufactured ... Convex was formed in 1982 by Bob Paluck and Steve Wallach in Richardson, Texas . It was originally ... lower performance, but with a much better price performance ratio . In order to lower costs, the Convex ... of Unix known initially as Convex Unix then later as ConvexOS due to trademark and licensing ... per CPU 100 MFLOPS peak for single precision . It was Convex s most successful product. The C2 ... offered, with 50 to 240 MFLOPS per CPU. However, the C3 and the Convex business model were overtaken ... Convex was the first vendor to ship a GaAs based product, they were losing money. In 1994, Convex ... for parallel computing could draw in customers. But the type of customers Convex attracted believed ... performance problems which could not easily be fixed. Eventually, Convex established a working ... HP UX to the platform and sell the platform as standalone servers. In 1995, Hewlett Packard bought Convex. HP sold Convex Exemplar machines under the S Class MP and X Class CC NUMA titles, and later incorporated ... According to most former employees, Convex was a very fun place at which to work. For some time, there were beer parties every Friday, and an annual Convex Beach Party where a truck load of sand would ... dedication and hard work, but also emphasized fun and creativity. Convex had an unusually thorough ... hours ensuring Convex s success. The culture was one of creativity. Especially in the first few years ... for the customer today? Dubious date July 2011 Convex lasted longer than most minisupercomputer companies, and to celebrate this and more so to remind themselves of the difficulties of the market, Convex ... Anderson Forest title CONVEX WANTS TO BE A FULL FLEDGED HEAVYWEIGHT url http www.businessweek.com archives 1991 b3210058.arc.htm accessdate 2009 05 29 ref Ex employees of Convex jokingly refer to themselves as ex cons . There is a http www.ex convex.org mailing list of Convex ex employees, as well as frequent ... more details
Refimprove date May 2010 No footnotes date May 2010 Artwork image file Convex and Concave.JPG title Convex and Concave artist M. C. Escher year 1955 type Lithography lithograph height 27.5 width 33.5 Italic title Convex and Concave is a Lithography lithograph print by the Netherlands Dutch artist M. C. Escher , first printed in March 1955. It depicts an ornate architectural structure with many stairs, pillars and other shapes. The relative aspects of the objects in the image are distorted in such a way that many of the structure s features can be seen as both convex shapes and concave impressions. This is a very good example of Escher s mastery in creating illusion of Impossible Architectures . The window s, roads, stairs and other shapes can be perceived as opening out in seemingly impossible ways and positions. The trick of using the cubes that appear as the motif in the Flag on right half of this print is easily identified. One can view these features as concave by viewing the image upside down. Note that all additional elements and decoration on the left are consistent with a viewpoint from above, while those on the right with a viewpoint from below hiding half the image makes it very easy to switch between convex and concave. See also Printmaking Sources Locher, J.L. 2000 . The Magic of M. C. Escher . Harry N. Abrams, Inc. ISBN 0 8109 6720 0. M. C. Escher Category Works by M. C. Escher Category 1955 works printmaking stub he ... more details
Contents mostly taken from Legendre transformation . In mathematics , convex conjugation is a generalization ... f X to mathbb R cup infty math taking values on the extended real number line the convex conjugate ... of the convex hull of the function s Epigraph mathematics epigraph in terms of its supporting hyperplane s. http maze5.net ?page id 733 Examples The convex conjugate of an affine function math f x left ... cases b, & x a infty, & x ne a. end cases math The convex conjugate of a power function math f x ... 1 p tfrac 1 q 1. math The convex conjugate of the absolute value function math f x left x right ... math The convex conjugate of the exponential function math f x , e x math is math f star left x right begin cases x ln x x , & x 0 0 , & x 0 infty , & x 0. end cases math Convex conjugate and Legendre transform of the exponential function agree except that the domain mathematics domain of the convex ... E left min x,X right math has the convex conjugate math begin align f star p int 0 p F 1 q , dq ... function f in particular, math f text inc f math for &fnof nondecreasing. Properties The convex conjugate of a closed convex function is again a closed convex function. The convex conjugate of a polyhedral convex function a convex function with Polyhedron polyhedral epigraph mathematics epigraph is again a polyhedral convex function. Convex conjugation is Order theory order reversing if math ... The convex conjugate of a function is always lower semi continuous . The biconjugate math f math the convex conjugate of the convex conjugate is also the closed convex hull , i.e. the largest lower semi continuous convex function with math f le f math . For Proper convex function proper functions f , f f sup sup if and only if f is convex and lower semi continuous. Fenchel s inequality For any function f and its convex conjugate f sup sup Fenchel s inequality also known as the Fenchel Young inequality ... x f star p . math The following lines were taken from Legendre transformation . Scaling properties Convex ... more details
object. Discrete geometry has large overlap with convexgeometry and computational geometry , and is closely related to subjects such as finite geometry , combinatorial optimization , digital geometry , discrete differential geometry , geometric graph theory , toric geometry , and combinatorial topology ... geometry Polyhedron Polyhedra and polytope s Polyhedral combinatorics Convex lattice polytope Lattice ... Gruber, Peter M. title Convex and Discrete Geometry publisher Springer location Berlin year 2007 isbn ... redirect3 Combinatorial geometry The term combinatorial geometry is also used in the theory of matroid s to refer to a simple matroid , especially in older texts Discrete geometry and combinatorial geometry are branches of geometry that study Combinatorics combinatorial properties and constructive methods of discrete mathematics discrete geometric objects. Most questions in discrete geometry involve ... geometry point s, line geometry lines , plane geometry plane s, circle s, sphere s, polygon ... Kepler Kepler and Augustin Louis Cauchy Cauchy , modern discrete geometry has its origins ... Thue , projective configuration s by Reye and Ernst Steinitz Steinitz , the geometry of numbers by Minkowski ... Graphs Geometry Structural rigidity and flexibility Cauchy s theorem geometry Cauchy s theorem Flexible polyhedron Flexible polyhedra Incidence structure s Configuration geometry Configurations ... group s Triangle group s Digital geometry Discrete differential geometry Geometric set partitioning and transversals See also Discrete and Computational Geometry Discrete mathematics Paul Erd s References cite book author Bezdek, Andr s Kuperberg, W. title Discrete geometry in honor of W ... 3 cite book author K roly Bezdek Bezdek, K roly title Classical Topics in Discrete Geometry publisher ... in discrete geometry publisher Springer location Berlin year 2005 isbn 0 387 23815 8 cite book last1 Pach first1 J nos authorlink1 J nos Pach last2 Agarwal first2 Pankaj K. title Combinatorial geometry ... more details
Optimal convex decompositions title Computational Geometry year 1985 editor first G.T. editor last Toussaint ...File Pentagon.svg right thumb 150px An example of a convex polygon a regular polygon regular pentagon In geometry , a polygon can be either convex or concave non convex . Convex polygons A convex polygon is a simple polygon whose Interior topology interior is a convex set . ref http www.mathopenref.com polygonconvex.html Definition and properties of convex polygons with interactive animation. ref The following properties of a simple polygon are all equivalent to convexity Every internal angle is less than or equal to 180 Degree angle degrees . Every line segment between two vertex geometry vertices remains inside or on the boundary of the polygon. A simple polygon is strictly convex if every internal angle is strictly less than 180 degrees. Equivalently, a polygon is strictly convex if every line segment between two nonadjacent vertices of the polygon is strictly interior to the polygon except at its endpoints. Every nondegenerate triangle is strictly convex. Concave or non convex polygons File Simple polygon.svg thumb 150px An example of a concave polygon. A simple polygon that is not convex ... Theory Into Practice isbn 0763722502 page 130 . ref non convex ref Citation last Leff first Lawrence title Let s Review Geometry year 2008 publisher Barron s Educational Series location Hauppauge .... ref It is always possible to cut a concave polygon into a set of convex polygons. clarify date September 2011 A polynomial time algorithm for finding a decomposition into as few convex polygons as possible ... . ref See also Convex hull References references External links mathworld urlname ConvexPolygon title Convex polygon mathworld urlname ConcavePolygon title Concave polygon http www.rustycode.com tutorials convex.html Category Convexgeometry Category Polygons ar bs Konveksni poligon ca Pol gon convex et Kumer hulknurk es Pol gono convexo eo Konveksa plurlatero eu Poligono ganbil ... more details
A convex lattice polytope also called Z polyhedron or Z polytope is a geometry geometric object playing an important role in discrete geometry and combinatorial commutative algebra . It is a polytope in a Euclidean space R sup n sup which is a convex hull of finitely many points in the integer lattice Z sup n sup &sub R sup n sup . Such objects are prominently featured in the theory of toric variety toric varieties , where they correspond to polarized projective toric varieties. Examples An n dimensional simplex &Delta in R sup n sup is the convex hull of n 1 points that do not lie on a single affine hyperplane. The simplex is a convex lattice polytope if and only if the vertices have integral coordinates. The corresponding toric variety is the n dimensional projective space P sup n sup . The unit cube in R sup n sup , whose vertices are the 2 sup n sup points all of whose coordinates are 0 or 1 , is a convex lattice polytope. The corresponding toric variety is the Segre embedding of the n fold product of the projective line P sup 1 sup . In the special case of two dimensional convex lattice polytopes in R sup 2 sup , they are also known as convex lattice polygons . In algebraic geometry, an important instance of lattice polytopes called the Newton polytopes are the convex hulls of the set math A math which consists of all the exponent vectors appearing in a collection of monomials. For example, consider the polynomial of the form math axy bx 2 cy 5 d math with math a,b,c,d neq 0 math has a lattice equal to the triangle math rm conv 1,1 , 2,0 , 0,5 , 0,0 . math See also Normal polytope Pick s theorem Ehrhart polynomial Integer points in convex polyhedra References Ezra Miller, Bernd Sturmfels , Combinatorial commutative algebra . Graduate Texts in Mathematics, 227. Springer Verlag, New York, 2005. xiv 417 pp. ISBN 0 387 22356 8 geometry stub Category Polytopes Category Lattice points ... more details
In number theory , the geometry of numbers studies convex body convex bodies and lattice group lattice s integer vectors in n dimensional space. The geometry of numbers was initiated by harvs txt authorlink Hermann Minkowski first Hermann last Minkowski year 1910 . The geometry of numbers has a close ... edition 2006. P. M. Gruber, Convex and discrete geometry, Springer Verlag, New York, 2007. P. M. Gruber, J. M. Wills editors , Handbook of convexgeometry. Vol. A. B, North Holland, Amsterdam, 1993 ... order lattice in n dimensional Euclidean space R sup n sup and K is a convex centrally symmetric ... 2 n vol R n Gamma . math Later research in the geometry of numbers In 1930 1960 research on the geometry ... theories that enumerate the lattice points in some convex bodies. ref Gr tschel et alia, Lov sz et ... Siegel s lemma volume mathematics determinant Parallelepiped In the geometry of numbers, the subspace ... on functional analysis main normed vector space see also Banach space F space Minkowski s geometry ... convex bodies induce normed space norms in finite dimensional vector spaces. Minkwoski s theorem was generalized ... convex sets that are closed and bounded generate the topology of a Banach space . ref For Kolmogorov ... set s and other convex set non convex set s. ref Kalton et alia. Gardner ref References references ... cite book author Enrico Bombieri and Walter Gubler title Heights in Diophantine Geometry publisher Cambridge U. P. year 2006 J. W. S. Cassels . An Introduction to the Geometry of Numbers . Springer ... , Springer, 1988 cite book author Hancock, Harris title Development of the Minkowski Geometry of Numbers ... 0 521 27585 7 mr 0808777 C. G. Lekkerkererker. Geometry of Numbers . Wolters Noordhoff, North Holland ... title Geometry of numbers first A.V. last Malyshev Citation last1 Minkowski first1 Hermann author1 ... Siegel title Lectures on the Geometry of Numbers year 1989 publisher Springer Verlag Rolf Schneider, Convex bodies the Brunn Minkowski theory, Cambridge University Press, Cambridge, 1993. Anthony C. Thompson ... more details
Image Orthogonal convex hull.svg thumb The orthogonal convex hull of a point set In Euclidean geometry , a set math K subset R n math is defined to be orthogonally convex if, for every line L that is parallel ..., a point, or a single interval. Unlike ordinary convex set s, an orthogonally convex set is not necessarily connectedness connected . The orthogonal convex hull of a set math S subset R n math is the intersection of all connected orthogonally convex supersets of S . These definitions are made by analogy with the classical theory of convexity, in which K is convex set convex if, for every line L ... the lines for which this property is required to hold, so every convex set is orthogonally convex but not vice versa. For the same reason, the orthogonal convex hull itself is a subset of the convex hull of the same point set. A point p belongs to the orthogonal convex hull of S if and only if each ... convex hull is also known as the rectilinear convex hull , or the x y convex hull . Example The figure shows a set of 16 points in the plane and the orthogonal convex hull of these points. As can be seen in the figure, the orthogonal convex hull is a polygon with some degenerate edges connecting ... convex hull edges are horizontal or vertical. In this example, the orthogonal convex hull is connected. Algorithms Several authors have studied algorithms for constructing orthogonal convex ... Wood 1984 harvtxt Karlsson Overmars 1988 . By the results of these authors, the orthogonal convex ... orthogonal convexity to restricted orientation convexity , in which a set K is defined to be convex ... is closely related to the orthogonal convex hull. If a finite point set in the plane has a connected orthogonal convex hull, that hull is the tight span for the Manhattan distance on the point set. However ... convexity and orthogonal visibility geometry visibility . References citation last1 Fink first1 ... last2 Wood first2 Derick doi 10.1007 BF01237603 journal Journal of Geometry pages 99 120 title Generalized ... more details
Elementary geometry Category Convexgeometry Category Polyhedra ar ca Cara superf cie ...Image Tile 4,4.svg thumb Square tiling four square geometry square faces per vertex Image hexahedron.png thumb Cube three square geometry square faces per vertex In geometry , a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the square geometry square s that bound a cube is a face of the cube. The suffix hedron is derived from the Greek word hedra which means face . Sometimes, in the case of a pyramid , the term face is understood to exclude the base. The two dimensional polygons that bound higher dimensional polytopes are also commonly called faces . Formally, however, a face is any of the lower dimensional boundaries of the polytope, more specifically called an n face . Formal definition In convexgeometry , a face of a polytope P is the intersection of any supporting hyperplane of P and P . From this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set. For example, a polyhedron R sup 3 sup is entirely on one hyperplane of R sup 4 b . If R sup 4 sup were spacetime, the hyperplane at nowrap t 0 supports and contains the entire polyhedron. Thus, by the formal definition, the polyhedron is a face of itself. All of the following are the n faces of a 4 polytope 4 dimensional polytope 4 face the 4 dimensional 4 polytope itself 3 face any 3 dimensional cell geometry cell 2 face any 2 dimensional polygonal face using the common definition of face 1 face any 1 dimensional edge geometry edge 0 face any 0 dimensional vertex geometry vertex the empty set. Facets If the polytope lies in n dimensions, a face in the n 1 dimension is called a Facet mathematics facet . For example, a cell of a polychoron is a facet, a face of a polyhedron is a facet, an edge of a polygon is a facet, etc. A face in the n 2 dimension is called a Ridge geometry ridge . See also Euler characteristic External ... more details
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