In economics , convexpreferences 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 marginal utility The law of diminishing marginal utility the law of 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 ... a positive amount of each bundle is viewed as being better than the third bundle. A set of Convex function convex shaped indifference curve s displays convexpreferences Given a convex indifference ... curve is a convex set . Convexpreferences with their associated convex indifference mapping arise from Quasi convex function quasi concave utility functions, although these are not necessary for the analysis of preferences. References Hal R. Varian Intermediate Microeconomics A Modern Approach ... is strictly convex if for any math x, y, z in X math where math y succeq x math , math z succeq x math ... the preference ordering P is strictly convex if for any two distinct goods bundles that are each ... 507340 9 See also Convex function Level set Quasi convex function Semi continuous function Shapley Folkman ... 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 plane graph , a plane graph with convex faces Convex optimization Economics Convexpreferences , a preference relation with convex upper contour sets Finance Bond convexity , a measure of the sensitivity of the price of a bond to changes in interest rates 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 See also Concave disambiguation , the opposite of convex List of convexity topics Obtuse angle disambig cs Konvexn de Konvex it Convesso ja ro Convexitate fi Konveksisuus sv Konvex ... more details
Orphan date February 2009 Expert subject Economics date November 2008 GHH preferences short for Greenwood Hercowitz Huffman preferences , refer to an economic formula developed by Jeremy Greenwood, Zvi Hercowitz, and Gregory Huffman, in their 1988 paper Investment, Capacity Utilization, and the Real Business Cycle . It describes the macroeconomics macroeconomic impact of technological changes that affect productivity. GHH preferences have Gorman form . References http www.econ.rochester.edu Faculty GreenwoodPapers ghh88.pdf Investment, Capacity Utilization, and the Real Business Cycle Jeremy Greenwood archive http www.econterms.com glossary.cgi?action Search &query GHH preference GHH preference at the Glossary of Economics Research Category Macroeconomics econ stub ... more details
Unreferenced date August 2008 Lexicographic preferences lexicographical order based on the order of amount of each good describe comparative preferences where an agent economics economic agent infinitely prefers one good X to another Y . Thus if offered several bundles of goods, the agent will choose the bundle that offers the most X, no matter how much Y there is. Only when there is a tie of Xs between bundles will the agent start comparing Ys. For example, if for a given bundle X Y Z an agent orders his preferences according to the rule X Y Z, then the bundles 5 3 3 , 5 1 6 , 3,5,3 would be ordered, from most to least preferred 5 3 3 5 1 6 3 5 3 Even though the first option contains fewer total goods than the second option, it is preferred because it has more Y. Note that the number of X s is the same, and so the agent is comparing Y s. Even though the third option has the same total goods as the first option, the first option is still preferred. Even though the third option has far more Y than the second option, the second option is still preferred because it has slightly more X. Implications If all agents have the same lexicographic preferences, then general equilibrium cannot exist because agents won t sell to each other as long as price of the less preferred is more than 0 number zero . But if the price of the less wanted is zero, then all agents want an infinite amount of the good. Equilibrium cannot be attained. Lexicographic preferences can still exist with general equilibrium. For example, Different people have different bundles of lexicographic preferences such that different individuals value items in different orders. Some, but not all people have lexicographic preferences. Lexicographic preferences extend only to a certain quantity of the good. Lexicographic preferences are the classical example of rational preferences that are not representable by a ml Utility Utility functions utility function , if amounts can be any non negative real value. If there were ... more details
In economics, an agent s preferences are said to be weakly monotonic if, given a consumption bundle math x math , the agent prefers all consumption bundles math y math that have more of every good. That is, math y gg x math implies math y succ x math . An agent s preferences are said to be strongly monotonic if, given a consumption bundle math x math , the agent prefers all consumption bundles math y math that have more of at least one good. That is, math y geq x math and math y neq x math imply math y succ x math . This definition defines monotonic increasing preferences. Monotonic decreasing preferences can often be defined to be compatible with this definition. For instance, an agent s preferences for pollution may be monotonic decreasing less pollution is better . In this case, the agent s preferences for lack of pollution are monotonic increasing. Much of consumer theory relies on a weaker assumption, local nonsatiation . An example of preferences which are weakly monotonic but not strongly monotonic are those represented by a Leontief Utilities Leontief utility function . References Andreu Mas Colell Mas Colell, Andreu , Whinston, Michael D., Green, Jerry R. Microeconomic Theory. Oxford University Press. 1995. See also Monotonic function Monotonicity in calculus and analysis Strict Category Microeconomics Category Consumer theory microeconomics stub uk ... more details
unreferenced date May 2010 Homothetic preferences is a subset of preferences used in economics when analyzing the demand for Consumption economics consumption of various goods and services . In mathematics, a function of two or more arguments is homothetic if all ratios of its first partial derivatives depend only on the ratios of the arguments, not their levels see homothetic transformation . In economics, a model with competitive consumers or producers optimizing subject to homothetic utility or production functions, will imply that ratios of goods demanded depend only on relative prices, not on income or scale. Intratemporally homothetic preferences This assumption on individual preferences assures that decision makers with different incomes but facing the same prices will demand goods in the same proportions. Intertemporally homothetic preferences Models of modern macroeconomics and public finance often assume the constant relative risk aversion form for within period utility also called the power utility power or isoelastic form . The reason is that, in combination with additivity over time, this gives homothetic intertemporal preferences and this homotheticity is of considerable analytic convenience for example, it allows for the analysis of steady states in growth models . These assumptions imply that the elasticity of intertemporal substitution , and its inverse, risk aversion fluctuation risk aversion , are constant rich and poor decision makers are equally averse to proportional fluctuations in consumption. This may have significant implications, for example when evaluating the Welfare cost of business cycles costs of business cycles or evaluating a policy change in a dynamic general equilibrium model with heterogeneous agents. economics stub DEFAULTSORT Homothetic Preferences Category Utility Category Decision theory ... more details
Social preferences are a type of preference studied in behavioral economics behavioral and experimental economics experimental economics and social psychology , including interpersonal altruism , Fair division fairness , reciprocity , and inequity aversion . The term social preferences incorporates obstreperous esp. the Fehr Schmidt inequity aversion model and non obstreperous e.g., vulnerability based theories. Much of the recent evidence used to test society ideas and models has come from economics experiments. However, social preferences also matter outside the laboratory ref Gary Becker, The Economics of Discrimination ref ref Enzo Dietrich, Neighbors are Negative, Quarterly Journal of Economics ref . References references socio stub econ stub Category Sociology Category Social psychology Category Social sciences ... more details
Unreferenced date June 2008 Endogenous preferences are preference s that cannot be taken as given, but are affected by individual internal responses to the external state of affairs. They are interdependent, in part determined by social institutions, marketed advertisement, and subject to learning experience and observation and habit formation past experience . See also Acquired taste economics stub Category Economics ... more details
noreferences date November 2007 Infobox Software name System Preferences logo Image System Preferences icon.png 64px screenshot Image System Preferences Snow Leopard .png 220px caption System Preferences application in Mac OS X Snow Leopard . developer Apple Inc. latest release version 7.0 7.0 latest release date August 28, 2009 operating system Mac OS X genre Computer configuration Settings license Proprietary software Proprietary Deleted image removed Image Preference Pane 10.4.png 90px thumb right Preference Pane from Mac OS X 10.4. Commented out because image was deleted Image Screen Saver 10.4.png 90px thumb right Screen Saver from Mac OS X 10.4. System Preferences is an Application software application included with the Mac OS X operating system that allows users to modify various system settings which are divided into separate preference pane s. The System Preferences application was introduced in the first version of Mac OS X to replace the control panel Mac OS control panel that was included in previous versions of the Mac OS Mac operating system . Overview History Before the release of Mac OS X in 2001, users modified system settings using control panels. Control panels, unlike the preference panes found in System Preferences, were separate applications that were accessed ... panes are not applications but subsections of the System Preferences application. By default, System Preferences organizes preference panes into several categories. In the latest version of System Preferences, included with Mac OS X v10.6 , these categories are Personal , Hardware , Internet .... Users can also choose to sort preference panes alphabetically. Originally, System Preferences ... Preferences. This was replaced by a single Dashboard & Expos pane in Mac OS X v10.4, which introduced ... settings. MobileMe used to set preferences for the user s MobileMe account and iDisk. computer network ... only software made by Apple Inc. it Preferenze di Sistema ja uk System Preferences ... more details
In the Color psychology psychology of color , color preferences are the tendency for an individual or a group to prefer some color s over others, including a favorite color . See also Car colour popularity Car color popularity Color psychology Color theory Color vision L scher color test Further reading expand further Citation last Crozier year 1999 first W. Ray title The meanings of colour preferences among hues journal Pigment & Resin Technology volume 28 issue 1 pages 6 14 doi 10.1108 03699429910252315 Citation last1 Ellis last2 Ficek date December 2001 first1 Lee first2 Christopher title Color preferences according to gender and sexual orientation journal Personality and Individual Differences volume 31 issue 8 pages 1375 1379 doi 10.1016 S0191 8869 00 00231 2 Citation last1 Grossman last2 Wisenblit year 1999 first1 Randi first2 Joseph Z. Priluck title What we know about consumers color choices journal Journal of Marketing Practice Applied Marketing Science volume 5 issue 3 pages 78 88 doi 10.1108 EUM0000000004565 Citation last1 Madden last2 Hewett last3 Roth year 2000 first1 Thomas J. first2 Kelly first3 Martin S. title Managing Images in Different Cultures A Cross National Study of Color Meanings and Preferences journal Journal of International Marketing volume 8 issue 4 pages 90 107 doi 10.1509 jimk.8.4.90.19795 Citation last Morse date March 2008 first Janice M. title What s your favorite color? Reporting irrelevant demographics in qualitative research journal Qualitative Health Research volume 18 pages 299 300 doi 10.1177 1049732307310995 Citation last Saito first Miho date February 1996 title Comparative studies on color preference in Japan and other Asian regions, with special emphasis on the preference for white journal Color Research & Application volume 21 ... spontaneous color preferences are not due to adult like brightness variations journal Visual Neuroscience ... Prediction of infants spontaneous color preferences journal Vision Research volume 47 issue 10 pages ... more details
Legacy preferences or legacy admission is a type of preference given by educational institutions to certain applicants on the basis of their familial relationship to alumni of that institution. Students so admitted are referred to as legacies or legacy students . This preference is most common in American universities and colleges ref citation author Daniel Golden chapterurl http www.tcf.org publications education Legacy ch4.pdf title Affirmative Action for the Rich chapter Chapter 4 An Analytic Survey of Legacy Preference year 2010 ISBN 978 0870785184 ref and emerged after World War I, primarily in response to the resulting immigrant influx ref cite book title Color and Money How Rich White Kids Are Winning the War over College Affirmative Action author Peter G. Schmidt isbn 978 1403976017 year 2007 ref . The Ivy League institutions are estimated to admit 10 to 15 of each entering class based upon this factor. ref cite news url http www.economist.com displaystory.cfm?story id 2333345 work ... year 2010 date 2010 09 29 ref Legacy preferences in comparison to other programmes At some schools, legacy preferences have an effect on admissions comparable to other factors such as being a recruited ... children of alumni 160 ref citation title The Opportunity Cost of Admission Preferences at Elite Universities ... from alumni, critics argue that legacy preferences are a way to indirectly sell university placement ... Analysis of the Impact of Legacy Preferences on Alumni Giving at Top Universities year 2010 ... supporting or opposing both affirmative action and legacy preferences simultaneously. For example ... of all non academic preferences also point out that many European universities, including ... , do not use any racial, legacy, or athletic preferences in admissions decisions. There is also a legal argument against legacy preferences in government schools, which argues that they violate ... of Legacy Preferences in Public School Admissions , Washington University Law Review ... 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
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
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 . See also List of convexity topics References J. B. Hiriart Urruty, C. Lemar chal , Fundamentals of convex analysis, Springer Verlag, Berlin, 2001. R. T. Rockafellar , Convex analysis, Princeton University Press, Princeton, NJ, 1970. Reprint 1997. cite book last Singer first Ivan title Abstract convex analysis series Canadian Mathematical Society series of monographs and advanced texts publisher John Wiley & Sons, Inc. location New York year 1997 pages xxii 491 isbn 0 471 16015 6 id MR 1461544 J. Stoer, C. Witzgall, Convexity and optimization in finite dimensions. I, Springer, Berlin, 1970. cite book last Z linescu first C. title Convex analysis in general vector spaces publisher World Scientific Publishing Co., Inc River Edge, NJ, 2002 pages xx 367 isbn 981 238 067 1 id MR 1921556 Category Convex analysis Category Mathematical optimization Category Mathematical analysis Category Variational analysis mathanalysis stub fr Analyse convexe nl Convexe analyse ... more details
Convex geometry 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 ... branches of the mathematical discipline Convex and Discrete Geometry are General Convexity , Polytopes ... 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 bodies approximation by convex sets variants of convex sets star shaped, m, n convex, etc. Helly ..., volume mixed volumes and related topics inequalities and extremum problems convex functions and convex programs spherical and hyperbolic convexity The phrase convex geometry is also used in combinatorics as the name for an abstract model of convex sets based on antimatroid s. Historical note Convex geometry is a relatively young mathematical discipline. Although the first known contributions to convex ... Fenchel W. Fenchel gave a comprehensive survey of convex geometry in Euclidean space R sup n sup . Further development of convex geometry in the 20th century and its relations to numerous mathematical disciplines are summarized in the Handbook of convex geometry edited by P. M. Gruber and J. M. Wills. See also List of convexity topics References Expository articles on convex geometry K. Ball, An elementary introduction to modern convex geometry, in Flavors of Geometry, pp. 1 58, Math. Sci ... , 47 83. V. Klee, What is a convex set? Amer. Math. Monthly, Vol. 78 1971 , 616 631. Some books on convex .... English translation Theory of convex bodies, BCS Associates, Moscow, ID, 1987. R. J. Gardner, Geometric tomography, Cambridge University Press, New York, 1995. Second edition 2006. P. M. Gruber, Convex ... more details
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 ..., 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 if and only if the region in green above its graph of a function graph in blue is a convex set. Let ... spaces. A set mathematics set C in S is said to be convex if, for all x and y in C and all ... point on the line segment connecting x and y is in C . This implies that a convex set in a real ... 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 ... of constant width . Some examples of convex subsets of Euclidean space Euclidean 3 space are the Archimedean solid s and the Platonic solid s. The Kepler Poinsot polyhedra are examples of non convex sets. Properties If math S math is a convex set, for any math u 1,u 2, ldots,u r math in math S math ... is in math S math . A vector of this type is known as a convex combination of math u 1,u 2, ldots,u r math . Intersections and unions The collection of convex subsets of a vector space has the following ... more details
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. 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 negative and sum up to 1. All possible convex combinations will be within the convex hull of the given points. In fact, the collection of all such convex combinations of points in the set constitutes the set s convex hull ... space , a convex combination of these points is a point of the form math alpha 1x 1 alpha 2x 2 cdots ... alpha 1 alpha 2 cdots alpha n 1. math As a particular example, every convex combination of two points ... that are not closed under linear combinations but that 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 A convex combination ..., 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 a different ... instead the sum is explicitly divided from the linear combination. Affine combination s are like convex ... s theorem convex hull Conical combination convex hull Conical combination Nonnegative linear combination Simplex DEFAULTSORT Convex Combination Category Convex geometry Category Mathematical analysis Category Convex hulls es Combinaci n convexa fr Combinaison convexe it Combinazione convessa pl Kombinacja ... 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 or on any convex subset of some vector space is called convex , concave upwards , concave up or convex cup , if for any two points math x 1 math and math x 2 math ... 1 t f x 2 . math A function is called strictly convex if math f tx 1 1 t x 2 t f x 1 1 t f x 2 , math ... be defined over a convex set , otherwise the point math tx 1 1 t x 2 , math may not lie in the function ... convex. Pictorially, a function is called convex if the function lies below or on the straight line ... is used A function is convex if its epigraph mathematics epigraph the set of points lying on or above the graph of a function graph is a convex set . These two definitions are equivalent, i.e., one ... of the red line in the above drawing note also that the function R is symmetric in x,y . f is convex ... of convexity is quite useful to prove the following results. A convex function f defined ... is shown in the examples section . A function is midpoint convex on an interval C if math f left ... weaker than convexity. For example, a real valued Lebesgue measurable function that is midpoint convex will be convex. ref Sierpinski Theorem, Donoghue 1969 , http books.google.com books?id P30Y7daiGvQC&pg PA12 p. 12 ref In particular, a continuous function that is midpoint convex will be convex. A differentiable function of one variable is convex on an interval if and only if its derivative is monotonically non decreasing on that interval. If a function is differentiable and convex then it is also continuously differentiable . A continuously differentiable function of one variable is convex ... of f x . A twice differentiable function of one variable is convex on an interval if and only ... more details
Unreferenced date December 2009 In linear algebra , a convex cone is a subset of a vector space that is closure mathematics closed under linear combination s with positive coefficients. Image Convex cone illust.svg right thumb A convex cone light blue . Inside of it, the light red convex cone consists ... symbolize that the regions are infinite in extent. Definition A subset C of a vector space V is a convex ... vector space 0 are convex cones by this definition. Other examples are the set of all positive multiples ... is a positive scalar and x is an element of some convex set convex subset X of V . In particular ... gives an open resp. closed convex circular cone . Convex cones are closed under intersection ..., 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 ... 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 ... , in the definition 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 ... open or closed are convex cones. Moreover, any convex cone C that is not the whole space V must be contained in some closed half space H of V . In fact, a topologically closed convex cone is the intersection ... open convex cone. Salient convex cones and perfect half spaces A convex cone is said to be flat if it contains some nonzero vector x and its opposite x and salient otherwise. A blunt convex cone ... 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 Geometry es Cuerpo convexo ... more details
Convex optimization , a subfield of optimization mathematics mathematical optimization , studies the problem of minimizing convex function s. Given a real number real vector space math X 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 is to find ... of math mathcal X math and math f math makes the powerful tools of convex analysis applicable the Hahn ... that for linear programming , and effective computational methods. Convex minimization has applications ... optimal design , and finance . With recent improvements in computing and in optimization theory, convex minimization is nearly as straightforward as linear programming . Convex optimization has applications beyond minimizing convex functions. Convex optimization is useful also for some obviously maximizing concave functions and for the theory of maximizing convex functions The problem of maximizing a concave function can be re formulated equivalently as a problem of minimizing a convex function. Consider the restriction of a convex function to a compact set compact convex set Then, on that set ... s Convex Analysis states this maximum principle for extended real valued functions. ref Such results ... , and partial differential equation s. Theory The following statements are true about the convex minimization ... is convex. for each strictly convex function, if the function has a minimum, then the minimum is unique. These results are used by the theory of convex minimization along with geometric notions from ... lemma . Standard form Standard form is the usual and most intuitive form of describing a convex minimization problem. It consists of the following three parts A convex function math f x mathbb ... math g i x leq 0 math , where the functions math g i math are convex Equality constraints of the form ... in the form math h i x Ax b math , where math A math is a matrix and math b math is a vector. A convex ... more details
Image Extreme points illustration.png thumb right The convex hull of the red set contains also the blue convex set . In mathematics , the convex hull or convex envelope for a Set mathematics set of points X in a real number real vector space V is the minimal convex set containing X . In computational geometry , a basic problem is finding the convex hull for a given finite nonempty set of points in the plane mathematics plane . It is common to use the term convex hull for the boundary topology boundary of that set, which is a convex polygon , except in the degenerate case that points are collinear . The convex hull is then typically represented by a sequence of the vertices of the line segment ... thumb Convex hull elastic band analogy For planar objects , i.e., lying in the plane, the convex ... object when released, it will assume the shape of the required convex hull. It may seem natural to generalise ... surface in this case may not be the convex hull parts of the resulting surface may have negative curvature ... will spring back under tension to take the form of the convex hull of the points. 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 the characteristic properties of a closure ... idempotent Conv Conv S Conv S . Thus, the convex hull operator is a proper hull operator. Algebraic characterization Algebraically, the convex hull of X can be characterized as the set of all of the convex combination s of finite subset s of points from X that is, the set of points of the form ... satisfies either of the two definitions above. So the convex hull math H mathrm convex X math ... more details
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 ... 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 ... of bounded convex polytopes can be found in the article polyhedron . In the 2 dimensional case ... 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 ... A convex polytope may be defined in a number of ways, depending on what is more suitable for the problem at hand. Gr nbaum s definition is in terms of a convex set of points in space. Other important ... , as the convex hull of a set of points vertex representation . Vertex representation Convex hull In his book Convex polytopes , Gr nbaum defines a convex polytope as a compact convex set A set math K subset mathbb R d math where math mathbb R math is real space is convex if, for each pair ... within K . A compact convex set math K subset mathbb R d math is a polytope provided ext K the set ... 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 ... to their systems. The machines ran a BSD version of Unix known initially as Convex Unix then later ..., and rated at 50 MFLOPS peak for double precision per CPU 100 MFLOPS peak for single precision . It was Convex ..., the C3 and the Convex business model were overtaken by changes in the computer industry. The arrival ... a business in decline. By this time, even though Convex was the first vendor to ship a GaAs based product, they were losing money. In 1994, Convex introduced an entirely new design, known as the Exemplar ... of customers Convex attracted believed in Fortran and brute force rather than sophisticated ... be fixed. Eventually, Convex established a working partnership with HP s hardware and software divisions ... servers. In 1995, Hewlett Packard bought Convex. HP sold Convex Exemplar machines under the S Class ... was sold with the S and X Class products. Culture According to most former employees, Convex was a very ... Convex Beach Party where a truck load of sand would be dumped on the parking lot to simulate a beach ... fun and creativity. Convex had an unusually thorough interview process, which, for technical positions ... employee base who spent most of their waking hours ensuring Convex s success. The culture was one ..., extolling such slogans as What have you done for the customer today? Convex lasted longer than ... of the market, Convex had a graveyard of former competitor companies on its property. ref cite web author Stephanie 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 ... more details
Encyclopedia
top
