Convex geometry

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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

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 .

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