41. Glossary A convex body is, technically, a closed and bounded convex set with nonzero volume. convex geometry The study of convex shapes, usually in Euclidean space. http://www.math.ucdavis.edu/glossary.html

Glossary by Greg Kuperberg Index: A B C D ... Z A algebraic geometry Traditionally, the geometry of solutions in the complex numbers to polynomial equations. Modern algebraic geometry is also concerned with algebraic varieties, which are a generalization of such solution sets, as well as solutions in fields other than complex numbers, for example finite fields. algebraic topology The branch of topology concerned with homology and other algebraic models of topological spaces algebraic variety A space which is locally the solution locus to a set of polynomial equations. Algebraic varieties are for algebraic geometry topological spaces are for topology manifolds . However, algebraic varieties may also have complicated singular sets and may be parametrized with rings other than the complex numbers. (For the technical reason that the real numbers are not algebraically closed, one does not consider algebraic varieties over the real numbers in the straightforward sense.) alternating-sign matrix A matrix of 0's, 1's, and -1's such that, if the zeroes are deleted from any row or column, the remaining entries alternate in sign and begin and end with 1. almost complex manifold A manifold with the property that each tangent space has the structure of a complex vector space, but the complex structures are not necessarily compatible with true complex coordinates as they are for a complex manifold.

42. Roman Vershynin: Papers The main result represents a new connection between the local asymptotic convex geometry (study of sections of convex bodies) and the global asymptotic convex http://www.math.ucdavis.edu/~vershynin/papers/papers.html

Papers R.Vershynin, Frame expansions with erasures: an approach through the non-commutative operator theory, submitted ps pdf dvi tex In modern communication systems such as the Internet, random losses of information can be mitigated by oversampling the source. This is equivalent to expanding the source using overcomplete systems of vectors (frames), as opposed to the traditional basis expansions. Dependencies among the coefficients in frame expansions often allow for better performance comparing to bases under random losses of coefficients. We show that for any n-dimensional frame, any source can be linearly reconstructed from only n log n randomly chosen frame coefficients, with a small error and with high probability. Thus every frame expansion withstands random losses better (for worst case sources) than the orthogonal basis expansion, for which the n log n bound is attained. The proof reduces to M.Rudelson's selection theorem on random vectors in the isotropic position, which is based on the non-commutative Khinchine's inequality. R.Vershynin, Isoperimetry of waists and local versus global asymptotic convex geometries

43. Conference03 It is dedicated to questions of asymptotic geometric analysis, to the interactions between finite(high-)dimensional Banach spaces and convex geometry. http://www.math.uni-kiel.de/convex/

Banach Spaces and Convex Geometric Analysis April 6 - April 12, 2003 Mathematisches Seminar, Universität Kiel, Germany Mathematisches Seminar J. Lindenstrauss Landau Center Hebrew University Jerusalem The photos are now available. The conference is organized jointly by the Landau Center at the Hebrew University and the University of Kiel. It is dedicated to questions of asymptotic geometric analysis, to the interactions between finite-(high-)dimensional Banach spaces and convex geometry. The emphasis will be to compare recent results and ideas in both areas and to study the methods used, in particular those of analytic and probabilistic nature. Typical topics would include e.g. convex geometric inequalities, volume calculations, measure transportation, random constructions of convex bodies, Dvoretzky's theorem and its ramifications, symmetrizations of convex bodies, approximations of convex bodies, random matrices, related methods in discrete mathematics or questions of complexity. The program is available now .

44. Department Of Mathematics - University Of Idaho convex geometry and Discrete Optimization. Mark Nielsen (also see the web page on Dr. Nielsen s research interests). http://www.uidaho.edu/math/research/geometry.htm

Convex Geometry and Discrete Optimization Mark Nielsen (also see the web page on Dr. Nielsen's research interests

45. Dept. Of Math -Research - Mathematical Biology convex geometry and Discrete Optimization. Mark Nielsen. http://www.uidaho.edu/LS/Math/research/geometry.htm

Convex Geometry and Discrete Optimization Mark Nielsen

46. Recent Publications, Gideon Schechtman convex geometry and Local Theory of Normed Spaces. An ``isomorphic version of Dvoretzky s theorem (with VD Milman), CR Acad. Sci. http://www.wisdom.weizmann.ac.il/mathusers/gideon/pubsTopics/recentPubsByTopicLo

Convex Geometry and Local Theory of Normed Spaces An ``isomorphic" version of Dvoretzky's theorem (with V.D. Milman Global vs. local asymptotic theories of finite dimensional normed spaces (with V.D. Milman ), Duke Math. J. 90 (1997), 7393. An ``isomorphic" version of Dvoretzky's theorem II (with V.D. Milman ), Convex geometric analysis (Berkeley, CA, 1996), Math. Sci. Res. Inst. Publ., 34, Cambridge Univ. Press, Cambridge, (1999). Averages of norms and behavior of families of projective caps on the sphere (with A.E. Litvak and V.D. Milman Averages of norms and quasi-norms (with A.E. Litvak and V.D. Milman ), Math. Annalen 312 (1998), 95124. (with A. Zvavitch ), Math. Nachr. 227 (2001), 133142. Concentration, results and applications , in preparation for handbook, preliminary version available. Finite dimensional subspaces of $L_p$ (with W.B. Johnson ), Handbook of the Geometry of Banach Spaces, Vol 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 837870. (with J. Zinn

47. Abteilung Für Analysis - Research Main research. convex geometry, In convex geometry, geometric and analytic methods are used to study convex sets and convex functions. http://dmg.tuwien.ac.at/fg6/research.html

Research Forschungsgruppe Konvexe und Diskrete Geometrie 1040 Wien, Austria Main research Convex Geometry In Convex Geometry, geometric and analytic methods are used to study convex sets and convex functions. Therefore Convex Geometry is situated between Analysis and Geometry.  At the Department of Analysis especially the following problems are studied: Approximation of convex bodies by polytopes Properties of typical convex bodies in the sense of Baire categories Characterization of special convex bodies Valuations on the space of convex bodies Affine geometry of convex bodies Geometric Probabilities The Theory of Geometric Probabilities, Integral Geometry and Stochastic Geometry are situated between Geometry and Probability Theory. At the Department of Analysis especially the following problems are studied: Approximation of convex bodies by random polytopes Questions on the geometric structure of random polytopes Geometry of Numbers Geometry of Numbers forms a bridge between convexity, Diophantine approximation and the theory of quadratic forms. Today it is an independent problem-oriented field of mathematics having relations with coding theory, numerical integration, computational geometry and optimization. Geometry of Numbers has a long tradition in Vienna and at the Department of Analysis the following problems are studied: Diophantine approximation Products of inhomogeneous linear forms Inverting Minkowski's theorem on linear forms TU Wien Deutsche Version Homepage

48. Foundations Of Convex Geometry Home Catalogue Foundations of convex geometry. Foundations of convex geometry. WA Coppel. Published March 1998. 236 pages. Paperback ISBN 0521639700. http://books.cambridge.org/catalogue.asp?isbn=0521639700

49. CMIS Research - Image Analysis - Applications/Work Overview - Hyperspectral Imag scene). Fig. 2 Toy example of convex geometry model (M = 3) with noise endmembers lie at the vertices of the triangle. The leading http://www.cmis.csiro.au/iap/RecentProjects/hyspec_eg.htm

About CSIRO Doing Business Education Publications ... Image Analysis Application Areas Biotechnology Health Asset Monitoring Exploration ... Other Areas Skills Segmentation Feature Extraction Statistical Analysis Stereo Vision ... Staff Image Analysis Current Research In Hyperspectral Imaging What is Spectroscopy? What is Hyperspectral Imaging? What are the Major Processing and Analysis Challenges for Hyperspectral Image Data? How are These Problems Currently Addressed? (a) (b) Fig. 1: (a) 54 AVIRIS shortwave infrared images of Oatman, Arizona (courtesy of NASA JPL). (b) "Stackplot" of spectra at 6 pixels in the Oatman Image. Please click on the images for an enlarged view. Fig. 2: Toy example of convex geometry model (M = 3) with noise: endmembers lie at the vertices of the triangle The leading hyperspectral image analysis package, ENVI , has a method which finds the "pointiest" pixels (i.e. near vertices) using the "Pixel Purity Index". Clusters of such points are identified interactively as likely endmember clusters. More sophisticated methods include those of Craig (1994) which finds the simplex of minimum volume with a given number of vertices and completely enclosing the data "cloud"; and the N-FINDR algorithm of Winter (1999), which finds the simplex of maximum volume whose vertices are constrained to be a subset of the data points. N-FINDR is in commercial use. The Craig and Winter solutions for the toy example are shown in Fig. 3 (in pink and blue respectively). Note that Craig's solution is too large in the presence of noise, while Winter's will be too small if some materials in the scene are not represented by whole pixels.

50. Polytechnic University Department Of Mathematics: Instructional Team Erwin Lutwak Ph.D., Polytechnic Institute of Brooklyn convex geometry, geometric and analytic inequalities Phone (718) 2603366 Email elutwak@poly.edu Office http://www.math.poly.edu/people/instructional_team.phtml

Instructional Team Instructional Team Tenured Faculty Administrative Team Juan Carlos Alvarez Ph.D., Rutgers University Symplectic and contact geometry, integral geometry, Finsler manifolds, geometry of normed spaces Phone: Email: jalvarez@poly.edu Office: Keith Ball (on leave) Ph.D., Cambridge University Functional analysis, combinatorics, convexity David Chudnovsky Ph.D., Ukrainian Academy of Sciences Number theory, mathematical physics, large scale numerical computation Gregory Chudnovsky Ph.D., Ukrainian Academy of Sciences Number theory, mathematical physics, large scale numerical computation Jonathan Cornick Ph.D., Northern Illinois University Chomology of infinite groups, CW-complexes and homological algebra Phone: Email: jcornick@poly.edu Office: Jerome S. Epstein Ph.D., New York University Differential geometry and mathematical physics Phone: Email: jepstein@poly.edu Office: Zsuzsanna Gonye Ph.D., SUNY at Stony Brook Kleinian and Fuchsian groups, complex analysis Phone: Email: zgonye@poly.edu Office: Rachel Jacobovits M.S., Polytechnic University

51. The Finsler Geometry Newsletter - Home Page But, if we think of the great advances in convex geometry, the calculus of variations, integral geometry, the theory of metric spaces, and symplectic geometry http://www.math.poly.edu/research/finsler/

The Finsler Geometry Newsletter EDITORIAL PAGE AUTHOR INDEX INTRODUCTION PREPRINTS ... LINKS Created: Nov 18, 1999 Last update: News Welcome to the Finsler Geometry Newsletter. The aim of the Newsletter is to promote the interaction between researchers in convex, integral, metric, and symplectic geometry by providing them with a quick, accessible medium for communicating ideas, announcements, examples, counter-examples, and remarks. Criticisms and comments should be addressed to the webmaster The webmaster: Juan Carlos Alvarez New postings Volumes in normed and Finsler spaces On the perimeter and area of the unit disc Convex Bodies of Constant Width and Constant Brightness by Ralph Howard Remarks on magnetic flows and magnetic billiards, Finsler metrics and a magnetic analog of Hilbert's fourth problem by Serge Tabachnikov About Finsler Geometry Finsler manifolds, manifolds whose tangent spaces carry a norm that varies smoothly with the base point, were born prematurely in 1854 together with the Riemannian counterparts in Riemann's ground-breaking Habilitationsvortrag . I say prematurely because in 1854 Minkowski's work on normed spaces and convex bodies was still forty three years away, and thus not even the infinitesimal geometry on which Finsler manifolds are based was understood at the time. Apparently, Riemann did not know what to make of these 'more general class' of manifolds whose element of arc-length does not originate from a scalar product and, fatefully, put in a bad word for them:

52. Wlodek's Geometry Papers with convex sets, Chapter 3.3, vol. B, in Handbook of convex geometry, P. Gruber and J. Wills, Eds., NorthHolland 1993, 799-860. http://www.auburn.edu/~kuperwl/geometry.html

W. Kuperberg's Publications in Geometry W. Holsztynski and W. Kuperberg, On a property of tetrahedra, Wiadomosci Matematyczne (1962), 13-16 (in Polish), reviewed in Zentralblatt fur Mathematik 126, 369, English translation in Alabama Journal of Mathematics W. Kuperberg, Packing convex bodies in the plane with density greater than 3/4, Geometriae Dedicata W. Kuperberg, On minimum area quadrilaterals and triangles circumscribed about convex plane regions, Elemente der Mathematik W. Kuperberg, On packing the plane with congruent copies of a convex body, in Intuitive geometry, Coll. Math. Soc. Janos Bolyai 48, North-Holland, Amsterdam-Oxford-New York 1987, 317-329. W. Kuperberg, An inequality linking packing and covering densities of plane convex bodies, Geometriae Dedicata W. Kuperberg, Covering the plane with congruent copies of a convex body, Bulletin of the London Mathematical Society G. Kuperberg and W. Kuperberg, Double-lattice packings of convex bodies in the plane, A. Bezdek and W. Kuperberg, Maximum density space packing with congruent circular cylinders of infinite length

53. Convolution Inequalities In Convex Geometry Thematic Programme on Aymptotic Geometric Analysis Conference on Convexity and Asymptotic Theory of Normed Spaces. Convolution Inequalities in convex geometry. http://www.pims.math.ca/science/2002/aga/convexityvideos/ball/

Thematic Programme on Aymptotic Geometric Analysis: Conference on Convexity and Asymptotic Theory of Normed Spaces Keith Ball University College London Convolution Inequalities in Convex Geometry 9:00-9:45, Thursday, July 4, 2002 Math Annex 1100, UBC Watch or hear lecture now using the Real Player software. View abstract in PDF format. This talk was part of the Thematic Programme on Aymptotic Geometric Analysis: Conference on Convexity and Asymptotic Theory of Normed Spaces An file of this lecture can be downloaded here The complete archive of online lectures available from PIMS is available here Pacific Institute for the Mathematical Sciences Last Modified: Monday, 19-Aug-2002 03:45:14 PDT

54. AMS-UMI 2002 : Special Sessions 4) Analytic Aspects of convex geometry S. Campi (University of Modena) campi@unimo.it R. Gardner (Western Washington University) Richard.Gardner@wwu.edu E http://www.dm.unipi.it/~meet2002/italiano/session04.html

Sessioni Speciali - Abstracts "Analytic Aspects of Convex Geometry" S. Campi (University of Modena) campi@unimo.it R. Gardner (Western Washington University) Richard.Gardner@wwu.edu E. Lutwak (Polytechnic University Brooklyn) elutwak@duke.poly.edu A. Volcic (University of Trieste) volcic@univ.trieste.it Abstracts (tex format) Abstracts (ps format) Abstracts (dvi format) Sessioni Speciali

55. Introduction Convex is a Maple package for computations in rational convex geometry. The package provides functions for linear as well as affine convex geometry. http://www.mapleapps.com/powertools/convex/docs/intro.html

Contents Introduction Convex is a Maple package for computations in rational convex geometry. Here "rational" means that all coordinates must be rational numbers. The package provides functions for "linear" as well as "affine" convex geometry. In the affine setting, the basic objects are polyhedra, which are intersections of finitely many (affine) halfspaces. Polyhedra can also be described as the convex hull of finitely many points and rays. A bounded polyhedron is also called a polytope. In the Convex package, polyhedra are represented by the type POLYHEDRON . They may contain lines and may not be full-dimensional. The most important functions to define a POLYHEDRON are POLYHEDRON[convhull] and POLYHEDRON[intersection] The linear setting is based on cones, which are intersections of finitely many linear halfspaces (i.e., whose boundary contains the origin). They are generated by finitely many rays. In the Convex package, cones are represented by the type CONE . They may contain lines and may not be full-dimensional. A CONE can be created from either description with the functions CONE[poshull] and CONE[intersection] , respectively.

56. Maple Application Center Careers. Contact Us. Geometry, convex geometry, Solving and Displaying Inequalities, The Intersection of a Line and Cone, The Intersection of a Line and Cylinder, http://www.mapleapps.com/List.asp?CategoryID=11&Category=Geometry

57. DISPLAY AGENDAS A0344 Opening 15 , 1415 1515, Toric ring and discrete convex geometry (1) - 1h00 , W. Bruns Universität Osnabrück, Germany. 1515 1545, Break - 30 . 1545 1645, http://agenda2.ictp.trieste.it/cdsagenda5/full_display.php?ida=a0344

58. Geometric And Convex Combinatorics the solution of problems in integer programming have their origin in various fields of mathematics, such as Geometry of Numbers, convex geometry, Algebra, or http://www.math.uni-magdeburg.de/institute/imo/research/geometry_html/geometry.h

Next: References Geometric and convex combinatorics Methods for the solution of problems in integer programming have their origin in various fields of mathematics, such as Geometry of Numbers Convex Geometry Algebra , or Number Theory . The reason for this is the fact that the study of relations between discrete structures (lattices) and continuous sets (convex bodies, cones) is of fundamental importance for all of them. In this project we are trying to utilize current methods and results from the fields mentioned above for integer programming, and to contribute to a better understanding of lattice structures in connection with convex sets. The individual projects can be classified as follows: Geometry of Numbers Crepant Resolutions of Toric Singularities Test Sets in Integer Programming Packings and Coverings of Convex Bodies Geometry of Numbers Authors: Martin Henk, Robert Weismantel Cooperations: Support: Gerhard-Hess-Preis of the Deutschen Forschungsgemeinschaft, awarded to Robert Weismantel (We 1462/2-1) In 1896 Minkowski laid the foundation of what is today called the Geometry of Numbers , when he solved problems in number theory using geometric methods and interpretations. Today it is an independent field of research with close ties to other mathematical disciplines, for example coding theory and integer programming.

59. Semi-infinite Optimization Recent applications of semiinfinite optimization techniques to geometric extremal problems are opened up in the last years, first of all in convex geometry. http://www.math.uni-magdeburg.de/institute/imo/research/semiinfinite_html/semiin

Next: About this document ... Semi-infinite Optimization Authors: Friedrich Juhnke Staff Members: Cooperations: Semi-infinite Optimization deals with the problem of minimizing (maximizing) a real-valued objective function of a finite number of variables with respect to an (possibly and generally) infinite number of constraints. There is a great variety of (classical) applications of semi-infinite optimization, including problems in approximation theory (with respect to polyhedral norms), operation research, optimal control, boundary value problems and others. These applications and appealing theoretical properties of semi-infinite problems gave rise to intensive (and up to now undiminished) research activities in this field since its inceptive appearing in the 1960s. Recent applications of semi-infinite optimization techniques to geometric extremal problems are opened up in the last years, first of all in convex geometry. Describing an n-dimensional convex body by its Minkowski support function, there occur in a very natural way systems of (infinitely many) linear inequalities with a finite number of variables. Additionally, any inclusion of two convex bodies can equivalently be formulated by the inequality for all directions , where h k are the support functions of C K , respectively. So the feasible regions of extremum problems corresponding to coverings or embeddings in convex geometry can be described by semi-infinite systems and semi-infinite optimization techniques turn out to be an appropriate tool for handling them.

60. UCL > The Department Of Mathematics > The Department geometry of measures (as presented in Mattila Geometry of sets and measures in Euclidean spaces, Cambridge University Press 1995), convex geometry, ( see the http://www.ucl.ac.uk/Mathematics/Adverts/MarieCurie1.html

The Department Events Where to find us History and background Links ... Home Marie Curie Fellowships Studentships Available Geometric Real Analysis Applications are invited for Marie Curie fellowships to spend three months at University College London studying geometric real analysis. A description of the general mathematical area is given below. Up to 3 studentships in 2000 and 5 per year in the subsequent 3 years, are available. Keywords Combinatorial Geometry Probalistic Geometry Geometric Measure Theory Real and Convex Analysis Eligibility Applicants should be current PhD students working in geometric real analysis or on a project in any other area in which expertise in geometric real analysis would be helpful. They have to satisfy general eligibility criteria for Marie Curie Fellowships, which can be found on http://www.cordis.lu/improving. The main ones are that candidates must be citizens of the EU or an associated state, and must be registered for a PhD in an EU country other than the UK. Subject Area This broad field includes the geometry of measures (as presented in Mattila : Geometry of sets and measures in Euclidean spaces, Cambridge University Press 1995), convex geometry, ( see the introductory text , Ball elementary introduction to convex geometry, in Flavors of Geometry, Cambridge University Press 1997 , edited by Silvio Levy) , and the geometry of

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