41. Atlas: Convex Geometries: Recent Development Presented By K. Adaricheva In particular, the lattice of closed sets of any finite convex geometry is joinsemidistributive,and every finite join-semidistributive lattice can be http://atlas-conferences.com/cgi-bin/abstract/cake-48

Atlas Document # cake-48 65th Workshop on General Algebra, 18th Conference for Young Algebraists March 21-23, 2003 University of Potsdam Potsdam, Germany Organizers View Abstracts Conference Homepage Convex Geometries: recent development by K. Adaricheva Institute of Mathematics of SB RAS, Novosibirsk Convex Geometries: recent development Convex geometries are defined in combinatorics as the finite closure systems with the anti-exchange axiom . Via the lattices of closed sets they can be linked to the lattices with the unique irredundant decompositions that were studied in 40s by R.Dilworth. In recent paper by K.Adaricheva, V.Gorbunov and V.Tumanov "Join-semidistributive lattices and convex geometries'' (to appear in Adv.Math.) we discover a close connection of convex geometries with lattices satisfying the quasi-identity of join-semidistributivity: In particular, the lattice of closed sets of any finite convex geometry is join-semidistributive, and every finite join-semidistributive lattice can be embedded into the lattice of convex sets of some convex geometry. This also determines the place of the class of join-semidistributive lattices in the whole lattice hierarchy as a class that in some sense opposes to the class of modular lattices, the latter often being linked to the closure systems with the exchange-axiom The paper above introduces the general notion of a convex geometry as a (not necessarily finite) closure system with the anti-exchange axiom. This allows studying a wide class of closure systems that appear in different mathematical disciplines.

42. Semi-infinite Optimization Recent applications of semiinfinite optimization techniques to geometric extremalproblems 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.

43. Geometric And Convex Combinatorics the solution of problems in integer programming have their origin in various fieldsof 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.

44. Warwick Preprints By WS.Kendall 203, Computer algebra and stochastic calculus, 212, convex geometryand nonconfluent Gammamartingales I Tightness and strict convexity, http://www.warwick.ac.uk/statsdept/staff/WSK/ppt.html

45. 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

46. Polytechnic University Department Of Mathematics: Tenured Faculty Erwin Lutwak Ph.D., Polytechnic Institute of Brooklyn convex geometry, geometricand analytic inequalities Phone (718) 2603366 Email elutwak@poly.edu Office http://www.math.poly.edu/people/tenured_faculty.phtml

Tenured Faculty Instructional Team Tenured Faculty Administration 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: Kathryn Kuiken Ph.D., Polytechnic Institute of New York Group theory Phone: Email: kkuiken@poly.edu Office: Burton Lieberman Ph.D., New York University Differential equations; stochastic processes; statistics, sport science Phone: Email: blieber@poly.edu Office: Erwin Lutwak Ph.D., Polytechnic Institute of Brooklyn Convex geometry, geometric and analytic inequalities Phone: Email: elutwak@poly.edu Office: RH 305F Edward Y. Miller Ph.D., Harvard University Differential topology Phone: Email: emiller@poly.edu Office: Joel Rogers Ph.D., Massachusetts Institute of Technology Partial differential equations; fluid mechanics; numerical methods Email: rogers@poly.edu Lesley Sibner Ph.D., New York University Partial differential equations; global analysis Phone: Email: lsibner@poly.edu Office: Deane Yang Ph.D., Harvard University

47. MarieCurie1 geometry of measures (as presented in Mattila Geometry of sets and measures inEuclidean spaces, Cambridge University Press 1995), convex geometry, ( see the http://www.ucl.ac.uk/Mathematics/Adverts/MarieCurie1.html

48. IB Professor Imre Bárány, PhD, DSc. Tel 0207679-2836 E-mail barany@math.ucl.ac.uk.Research Interests convex geometry with Applications. http://www.ucl.ac.uk/Mathematics/staff/IB.html

49. Convex.nb All rights reserved. Please read this copyright notice. Introduction. In this notebookseveral useful calculations are set up for studies of convex geometry. http://www.mathphysics.com/convex/Convexnb.html

Calculations for Convex Bodies Introduction In this notebook several useful calculations are set up for studies of convex geometry. The original motivation for making the notebook was to study some properties of convex bodies of convex width, so some of the calculations are aimed at this problem. In particular, for that problem it is convenient to expand the important functions in Fourier series or spherical-harmonic series, depending on the dimension. Owing to these considerations, the notebook focuses on the relations among the following functions: The position function r (embedding the surface of a convex body K in R^2 or R^3 parametrically) The support function, which is H := r n , though usually expressed in terms of the angular coordinates of n , the normal vector to the supporting plane at r , rather than in terms of general parameters. The curvature. In this notebook, curvature will be described in terms of the principal radii of curvature; in 3 dimensions, in terms of the sum of the principal radii of curvature at r . As for the support function H, the curvature is often considered a function of the angular coordinates of

50. Maple Package `convex' convex a Maple package for convex geometry. Current Franz. This isa Maple package to facilitate computations in convex geometry. http://www.uni-konstanz.de/FuF/mathe/homepages/franz/convex/

convex - a Maple package for convex geometry Current version: Matthias Franz This is a Maple package to facilitate computations in convex geometry. Its provides functions to deal with rational polyhedral cones, polytopes, general polyhedra, faces of one of the above, and fans of (in principle) arbitrary dimension or size. For example, one can define cones and polyhedra as convex hulls of points, rays and lines or as intersections of halfspaces and hyperplanes, determine their elementary properties, like dimension, vertices, rays, and facet normals, plot polytopes up to dimension 3, dualise cones and polyhedra, test for containment, compatibility, and regularity, compute images and preimages under linear and affine maps, respectively, calculate convex hulls and intersections of cones and polyhedra, define Cartesian products, joins, and Minkowski sums, calculate the volume and surface of polytopes, compute in the face lattice of a cone or polyhedron, e.g., the supporting face of a list of points, the minimum or maximum of a list of faces

51. No Title LN Trefethen, D. Bau Numerical Linear Algebra, SIAM 1997. convex geometry Lecture4 h, 8 credit points; Tutorial 2 h, 2 credit points Prof. Wolfgang Weil. http://www.mathematik.uni-karlsruhe.de/~ipsm/abstract.html

INTERNATIONAL PROGRAMME IN MATHEMATICS ABSTRACTS OF THE CLASSES SUMMER SEMESTER 2003 (13 weeks) Boundary and Eigenvalue Problems for Partial Differential Equations Lecture: 4 h, 8 credit points; Tutorial: 2 h, 2 credit points Prof. Michael Plum Contents: Textbooks: Algebraic Geometry II Lecture: 4 h, 8 credit points; Tutorial: 2 h, 2 credit points Prof. Frank Herrlich Contents: This course gives an introduction to the modern point of view and techniques of algebraic geometry. In the place of the classical (point set) varieties we consider the set of prime ideals of a ring. Such spectra can under certain conditions be glued together to schemes . These are the central objects of the course; the quasiprojective varieties of the first course are contained in the theory as special cases. The most powerful general techniques to study schemes are sheaf theory and homological algebra. We shall introduce (quasicoherent) sheaves of modules on a scheme and also their cohomology groups. Specifically we shall study divisors and Picard groups on schemes, the sheaf

52. Sylvester's Four-Point Problem -- From MathWorld Weil, W. and Wieacker, J. Stochastic Geometry. Ch. 5.2 in Handbookof convex geometry (Ed. P. M. Gruber and J. M. Wills). Amsterdam http://mathworld.wolfram.com/SylvestersFour-PointProblem.html

Geometry Computational Geometry Convex Hulls Geometry ... Points Sylvester's Four-Point Problem Sylvester's four-point problem asks for the probability q R ) that four points chosen at random in a planar region R have a convex hull which is a quadrilateral (Sylvester 1865). Depending on the method chosen to pick points from the infinite plane, a number of different solutions are possible, prompting Sylvester to conclude "This problem does not admit of a determinate solution" (Sylvester 1865; Pfiefer 1989). For points selected from an open, convex subset of the plane having finite area , the probability is given by where is the expected area of a triangle over region R and A R ) is the area of region R . Note that is simply the value computed for an appropriate region, e.g., disk triangle picking triangle triangle picking square triangle picking , etc. P R ) can range between ) depending on the shape of the region, as first proved by Blaschke (Blaschke 1923, Peyerimhoff 1997). The following table gives the probabilities for various simple plane regions (Kendall and Moran 1963; Pfiefer 1989; Croft et al.

53. BioGeometry Recommended Literature convex geometry. Convex sets have been studied from a variety of angles.A big topic in this field is convex polytopes, and we recommend http://biogeometry.duke.edu/education/literature/geometry.html

Geometry Our sources of geometric thought and writing go back to ancient civilizations, most notably the Greek and the Chinese. As an introduction to this vast field, we recommend the book by Hilbert and Cohn-Vossen [ ], which illustrates the variety of geometric thinking within mathematics. We group our recommendations into six subareas. Elementary Geometry. This field studies simple geometric figures and their relationships. We mention Coxeter and Greitzer [ ] as an excellent source. We also recommend the book by Pedoe [ ], which contains a wealth of material on circles and spheres. Discrete Geometry. This area in geometry is dominated by the Hungarian school of thought, which includes work on packings and coverings as studied by Laszlo Fejes-Toth and combinatorial extremum problems as popularized by Paul Erdos. We recommend the text by Pach and Agarwal [ ], which discusses a broad range of problems and results in the area. Convex Geometry.

54. MaPhySto And StocLab Summer School On Stereology And Geometric Tomography Geometric tomography has connections with convex geometry, geometricprobing in robotics, computerized tomography, and other areas. http://www.maphysto.dk/events/S-and-GT2000/

MaPhySto and StocLab Summer School on Stereology and Geometric Tomography Sandbjerg Manor, 20-25 May, 2000 The aim of the summer school was to give an overview of modern stereology and its relation to geometric tomography, including both the mathematical and statistical theory and the practical applications. Content Stereology is the area of stochastics dealing with statistical inference about spatial structures from geometric samples of the structure such as two-dimensional sections and one-dimensional probes. The development of stereological methods involve the use of advanced mathematical tools, especially from geometric measure theory and integral geometry. Stereology is now in world-wide use in many areas of biology and medicine, most importantly in neuroscience and cancer grading. Other areas of application are geology, metallography and mineralogy. A disector consists of a reference plane and a look-up plane, a distance h apart. Illustration of the sampling relevant for estimation of length in R , based on projections on a vertical plane.

55. Convex Geometry Applied To Petri Nets State Space Size convex geometry Applied to Petri Nets State Space Size Estimationand Calculation of Traps, Siphons, and Invariants. Alexander http://pdv.cs.tu-berlin.de/~azi/texte/TB20006_info.html

Convex Geometry Applied to Petri Nets: State Space Size Estimation and Calculation of Traps, Siphons, and Invariants. Alexander Huck, Jörn Freiheit, Armin Zimmermann: Convex Geometry Applied to Petri Nets: State Space Size Estimation and Calculation of Traps, Siphons, and Invariants. Technical Report 2000-6, Technische Universität Berlin. Abstract The computation of Petri net properties - if solely based on its structural information - is less time and space consuming than methods based on a reachability graph analysis. In this paper we propose an improved algorithm for the generation of minimal invariants, traps, and siphons. Furthermore, a method for the efficient estimation of the state space size for Petri nets is presented, which uses the obtained structural properties. Both algorithms only require structural net information and are based on a new and improved implementation of the double description method. We show how this convex geometry technique can be successfully applied to Petri nets. Download will soon be available

56. Citation 5 BOARDMAN, J. 1993. Automating spectral unmixing of AVIRIS data using convex geometryconcepts. In the 4th JPL Airborne Geoscience Workshop (Washington, DC). http://www.acm.org/pubs/citations/journals/toms/1996-22-4/p469-barber/

home about feedback login Citation ACM Transactions on Mathematical Software (TOMS) archive Volume 22 , Issue 4 (December 1996) toc The quickhull algorithm for convex hulls Authors C. Bradford Barber Univ. of Minnesota, Minneapolis David P. Dobkin Princeton Univ., Princeton, NJ Hannu Huhdanpaa Configured Energy Systems, Inc., Plymouth, MN Publisher ACM Press New York, NY, USA Pages: 469 - 483 Periodical-Issue-Article Year of Publication: 1996 ISSN:0098-3500 http://doi.acm.org/10.1145/235815.235821 (Use this link to Bookmark this page) full text abstract references citings ... Review this Article Save to Binder BibTex Format FULL TEXT: Access Rules Click here to gain access to the Full Text! pdf 314 KB ABSTRACT REFERENCES Note: OCR errors may be found in this Reference List extracted from the full text article. ACM has opted to expose the complete List rather than only correct and linked references. 1 ALLEN, S. AND DUTTA, D. 1995. Determination and evaluation of support structures in layered manufacturing. J. Des. Manufactur. 5, 153-162.

57. Discrete Geometry - Publications Martin Henk a Series of Gorenstein Cyclic Quotient Singularities Admitting a Unique ProjectiveCrepant Resolution, to appaer in ``Combinatorial convex geometry and Toric http://fma2.math.uni-magdeburg.de/~henk/henk_pub.html

Martin Henk D-39106 Magdeburg, Germany Math. Reviews for Martin Henk I am happy to send (p)reprints of any of this to whoever is interested. henk@math.uni-magdeburg.de Publications On the representation of polyhedra by polynomial inequalities arXiv:math.MG/0203268 ), to appear in Discrete Comput. Geom. Dimitrios I. Dais and Martin Henk, On the Equations Defining Toric L.C.I. Singularities arXiv:math.AG/0204172 ), to appear in Trans. Amer. Math. Soc. Integral decomposition of polyhedra and some applications in mixed integer programming , Math. Prog. (B). Martin Henk, Successive Minima and Lattice Points , Rendi. Circ. Matematico Palermo, Serie II, Supppl. arXiv:math.MG/0204158 Martin Henk and Robert Weismantel, Diphantine Approximations and Integer Points of Cones , Combinatorica, On free planes in lattice ball packings , Bull. London Math. Soc. , no. 3, 2002, 284-290. Martin Henk and Annegret Wagler, Die Starke-Perfekte-Graphen-Vermutung , DMV-Mitteilungen Martin Henk and Chuaming Zong, Segments in Ball Packings , Mathematika, Ulrich Betke and Martin Henk

58. Richard J. Gardner's Home Page 1974, under the supervision of Professor CA Rogers, and the D.Sc degree from theUniversity of London in 1988 for work in measure theory and convex geometry. http://www.ac.wwu.edu/~gardner/

Richard Gardner Professor Richard.Gardner@wwu.edu Department of Mathematics College of Arts and Sciences Western Washington University Spring 2003 classes Math 312, Proofs in Elementary Analysis General Information Math 402/Math 502, (Introduction to) Abstract Algebra General Information Courses taught at Western Abstract Algebra : Math 401. Calculus : Math 124, Math 125, Math 224, Math 225, Math 226. Complex Analysis : Math 438/Math 538, Math 539. Discrete Mathematics and Combinatorics : Math 209, Math 566. Fourier Series and PDE's : Math 430/Math 530. Functional Analysis : Math 528. Geometric Tomography and Imaging : Math 397. Geometry : Math 560, Math 562. History of Mathematics : Math 420. Linear Algebra : Math 204, Math 304. Number Theory : Math 302. Optimization : Math-CS 335, Math-CS 435/Math 535, Math 570. Ordinary Differential Equations : Math 331, Math 432. Probability and Statistics : Math 341, Math 342, Math 441/Math 541. Real Analysis : Math 312, Math 421/Math 521, Math 422/Math 522, Math 527. Masters projects supervised at Western Olga Simek

59. Annex I To The Contract 3) Geometry of Banach spaces, convex geometry Develop the aspects of the geometryof Banach spaces already directly related to function theory and operator http://www.amsta.leeds.ac.uk/pure/analysis/network/annex1.html

60. Publications Translate this page Preprint ps et pdf. Entropy of the Grassmann manifold, convex geometryAnalysis, MSRI Publications, 34, (1998) 181-188. Version ps et pdf. http://www-math.univ-mlv.fr/~pajor/recherche/pub.htm

Ratios of Volumes and factorization through Illinois Jour. of Math. (1996), 91-107. Preprint ps et pdf Entropy of the Grassmann manifold , Convex Geometry Analysis, MSRI Publications, , (1998) 181-188. Version ps et pdf. The isotropy constants of the Schatten classes are bounded (1998), 773-783. Preprint ps et pdf (en collaboration avec A. Litwak et V. Milman) Proc. of AMS. (1999), 1499-1507. Preprint ps et pdf Metric entropy of convex hulls in Banach spaces , (avec B. Carl et I. Kyrezi) J. London Math. Soc. (1999) 871-896. Preprint ps et pdf The flatness theorem for non-symmetric convex bodies via the local theory of Banach spaces (en collaboration avec W. Banaszczyk, A. Litwak et S. Szarek) Mathematics of Operations research, No 3, (1999), 728-750. Preprint ps et pdf Entropy method in Asymptotic Convex Geometry (en collaboration avec V. Milman) C.R. Acad. Sci. Paris (1999), 303-308. Preprint ps et pdf Entropy and asymptotic geometry of non-symmetric convex bodies (en collaboration avec V. Milman) Advances in Math.

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