Convex geometry

The convex geometry (or convex geometry) is a branch of geometry. It was founded by Hermann Minkowski and treated the theory of convex sets in -dimensional real affine spaces or vector spaces. Minkowski developed his theory in his work, the geometry of numbers (Leipzig in 1896 and 1910).

The convex geometry has numerous references to other areas of mathematics such as number theory, functional analysis, or discrete mathematics.

Definition

A subset of a real - dimensional vector space is called convex if it contains with any two points A and B as well as all points between them, so the points of the segment AB. For each subset M of the real space exists its convex hull, which is the intersection of all convex sets containing M.

The convex hulls of finitely many points are called convex polyhedra or polytopes. Actual polytopes are those that are not in a true affine subspace. Classic examples are triangular, convex quadrilateral and parallelogram in the plane, tetrahedron, cube, octahedron, dodecahedron, icosahedron in three- simplex in arbitrary dimensions. You can polyhedra as unions of finitely many polytopes explain and build on this definition, the geometry of the polyhedron.

Selection of classical results of convex geometry

  • Set of barber
  • Selection set of Blaschke
  • Brunn - Minkowski inequality
  • Set of Carathéodory
  • Cauchy
  • Euler's formula
  • Set of Helly
  • Isoperimetric inequality
  • Set of young
  • Set of Kirchberger
  • Set of Krasnoselskii
  • Minkowski theorem
  • Minkowski shear lattice point set
  • Pick's theorem
  • Radon
Real number Curve of constant width Blaschke selection theorem Helly's theorem Jung's theorem Minkowski's theorem Pick's theorem Radon's theorem Krein–Milman theorem Choquet theory Wilhelm Blaschke Werner Fenchel Cambridge University Press Hugo Hadwiger American Mathematical Society Bibliographisches Institut
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