Convex Geometry

Convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas of mathematics: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming, probability theory, etc. According to the American Mathematical Society Subject Classification 2010, major branches of the mathematical discipline Convex and Discrete Geometry are: General Convexity, Polytopes and Polyhedra, Discrete Geometry. Further classification of General Convexity results in the following list:

• axiomatic and generalized convexity

• convex sets without dimension restrictions

• convex sets in topological vector spaces

• convex sets in 2 dimensions (including convex curves)

• convex sets in 3 dimensions (including convex surfaces)

• convex sets in n dimensions (including convex hypersurfaces)

• finite-dimensional Banach spaces

• random convex sets and integral geometry

• asymptotic theory of convex bodies

• approximation by convex sets

• variants of convex sets (star-shaped, (m, n)-convex, etc.)

• Helly-type theorems and geometric transversal theory

• other problems of combinatorial convexity

• length, area, volume

• mixed volumes and related topics

• inequalities and extremum problems

• convex functions and convex programs

• spherical and hyperbolic convexity

The phrase convex geometry is also used in combinatorics as the name for one of the abstract models of convex sets, one that is equivalent to antimatroids.