Convex geometry is the branch of geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

 studying convex set

Convex set

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...

s, mainly in Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

.
Convex sets occur naturally in many areas of mathematics: computational geometry

Computational geometry

Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational...

, convex analysis

Convex analysis

Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory....

, discrete geometry

Discrete geometry

Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles,...

, functional analysis

Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

, geometry of numbers

Geometry of numbers

In number theory, the geometry of numbers studies convex bodies and integer vectors in n-dimensional space. The geometry of numbers was initiated by ....

, integral geometry

Integral geometry

In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant transformations from the space of functions on one geometrical space to the...

, linear programming

Linear programming

Linear programming is a mathematical method for determining a way to achieve the best outcome in a given mathematical model for some list of requirements represented as linear relationships...

, probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

, etc. According to the American Mathematical Society

American Mathematical Society

The American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...

 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 volume
  Mixed volume
  In mathematics, more specifically, in convex geometry, the mixed volume is a way to associate a non-negative number to an n-tuple of convex bodies in the n-dimensional space...
  
  s 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

Combinatorics

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

 as the name for an abstract model of convex sets based on antimatroid

Antimatroid

In mathematics, an antimatroid is a formal system that describes processes in which a set is built up by including elements one at a time, and in which an element, once available for inclusion, remains available until it is included...

s.

Historical note

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

Euclid

Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

 and Archimedes

Archimedes

Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...

, it became an independent branch of mathematics at the turn of the 19th century, mainly due to the works of Hermann Brunn and Hermann Minkowski

Hermann Minkowski

Hermann Minkowski was a German mathematician of Ashkenazi Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.- Life and work :Hermann Minkowski was born...

 in dimensions two and three. A big part of their results was soon generalized to spaces of higher dimensions, and in 1934 T. Bonnesen and W. Fenchel

Werner Fenchel

Moritz Werner Fenchel was a mathematician known for his contributions to geometry and to optimization theory. Fenchel established the basic results of convex analysis and nonlinear optimization theory. Fenchel's monographs and lecture-notes were very influential also...

 gave a comprehensive survey of convex geometry in Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

Rⁿ. Further development of convex geometry in the 20th century and its relations to numerous mathematical disciplines are summarized in the Handbook of convex geometry edited by P. M. Gruber and J. M. Wills.