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Concept
Convex geometry
Summary
In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming, probability theory, game theory, etc.
According to the Mathematics Subject Classification MSC2010, the mathematical discipline Convex and Discrete Geometry includes three major branches:
general convexity
polytopes and polyhedra
discrete geometry
(though only portions of the latter two are included in convex geometry).
General convexity is further subdivided as follows:
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
valuations on convex bodies
inequalities and extremum problems
convex functions and convex programs
spherical and hyperbolic convexity
The term convex geometry is also used in combinatorics as an alternate name for an antimatroid, which is one of the abstract models of convex sets.
Convex geometry is a relatively young mathematical discipline. Although the first known contributions to convex geometry date back to antiquity and can be traced in the works of Euclid and Archimedes, it became an independent branch of mathematics at the turn of the 20th century, mainly due to the works of Hermann Brunn and Hermann Minkowski in dimensions two and three. A big part of their results was soon generalized to spaces of higher dimensions, and in 1934 T.
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