Convex geometry
The convex geometry (or convex geometry ) is a sub-area of geometry . It was founded by Hermann Minkowski and deals with the theory of convex sets in -dimensional real affine spaces or vector spaces . Minkowski developed his theory in his work Geometry of Numbers (Leipzig 1896 and 1910).
Convex geometry has numerous references to other areas of mathematics such as number theory , functional analysis , discrete mathematics or algebraic geometry ( toric geometry , tropical geometry ).
definition
A subset of a real -dimensional vector space is called convex if it contains two points each and also all points between them, i.e. the points of the segment . For every subset of real space there exists its convex hull , that is the intersection of all convex sets it contains.
The convex hulls of finitely many points are called convex polyhedra or polytopes . Real polytopes are those that do not lie in a real affine subspace . Classic examples are triangle, convex square and parallelogram in the plane, tetrahedron, cuboid, octahedron, dodecahedron, icosahedron in three-dimensional space, simplex in any dimensions. One can explain polyhedra as a union of finitely many polytopes and build the geometry of the polyhedron on this definition.
Selection of classic results of convex geometry
- Bárány theorem
- Set of barber
- Bieberbach's inequality
- Blaschke's selection sentence
- Brunn-Minkowski inequality
- Theorem of Carathéodory
- Cauchy's theorem
- Euler's polyhedron substitute
- Helly theorem
- Isoperimetric inequality
- Jung's theorem
- Lemma of Kakutani
- Kirchberger's theorem
- Krasnoselski's theorem
- Minkowski's theorem
- Minkovskian grid point theorem
- Theorem of Pick
- Theorem of radon
- Straszewicz's theorem
- Tverberg's theorem
- Motzkin theorem
- Tietze's theorem
- Hadwiger's theorem
Many of the sentences mentioned are only valid in infinite-dimensional spaces in a weakened form. See, for example, the Kerin-Milman theorem or Choquet theory .
literature
- Wilhelm Blaschke : circle and sphere . Verlag Walter de Gruyter , Berlin 1956.
- Wilhelm Blaschke: Collected Works. Vol. 3. Convex geometry. Edited by Werner Burau . Thales-Verlag, Essen 1985, ISBN 3-88908-203-3 .
- Tommy Bonnesen , Werner Fennel : Theory of convex bodies. Corrected reprint . Springer-Verlag, Berlin [a. a.] 1974, ISBN 3-540-06234-3 .
- Arne Brøndsted: An Introduction to Convex Polytopes . Springer-Verlag, New York [u. a.] 1983, ISBN 0-387-90722-X .
- WA Coppel: Foundations of Convex Geometry . Cambridge University Press, Cambridge 1998, ISBN 0-521-63970-0 .
- Peter M. Gruber: Convex and Discrete Geometry . Springer-Verlag, Berlin [a. a.] 2007, ISBN 978-3-540-71132-2 .
- Hugo Hadwiger : Old and new about convex bodies . Birkhäuser Verlag, Basel [u. a.] 1955.
- H. Hadwiger : Lectures on content, surface and isoperimetry (= the basic teachings of the mathematical sciences in individual presentations with special consideration of the areas of application . Volume 93 ). Springer-Verlag, Berlin ( inter alia) 1957 ( MR0102775 ).
- Isaak M. Jaglom , WG Boltjansky : Convex figures . German Science Publishing House, Berlin 1956.
- Victor L. Klee (Ed.): Convexity. Proceedings of the Seventh Symposium in Pure Mathematics of the American Mathematical Society, held at the University of Washington, Seattle, Washington, June 13-15, 1961 . American Mathematical Society , Providence RI 1963.
- Steven R. Lay: Convex Sets and their Applications . John Wiley & Sons, New York [u. a.] 1982, ISBN 0-471-09584-2 .
- Paul J. Kelly, Max L. Weiss: Geometry and Convexity . John Wiley & Sons, New York [u. a.] 1979, ISBN 0-471-04637-X .
- Kurt Leichtweiß : Convex sets . Springer-Verlag, Berlin [a. a.] 1980, ISBN 3-540-09071-1 .
- Jürg T. Marti : Convex Analysis . Birkhäuser Verlag, Basel [u. a.] 1977, ISBN 3-7643-0839-7 .
- Hermann Minkowski: Geometry of Numbers . Chelsea Publ., New York 1953 (Reprint of the 1896 edition).
- Athanase Papadopoulos: Metric Spaces, Convexity and Nonpositive Curvature (= IRMA Lectures in Mathematics and Theoretical Physics Vol. 6). European Mathematical Society, Zurich 2004, Second edition 2014, ISBN 978-3-03719-010-4 , p. 298.
- Frederick A. Valentine: Convex sets (= BI university pocket books. 402 / 402a). Bibliographical Institute, Mannheim 1968.
- Günter M. Ziegler : Lectures on Polytopes . Springer-Verlag, New York [u. a.] 1995, ISBN 0-387-94365-X .
Web link
- Ivan Izmestiev: Introduction to Convex Geometry. (PDF; 548 kB) script