In mathematics, convex geometry is the branch of

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

studying

convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...

s, mainly in

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...

. Convex sets occur naturally in many areas: computational geometry,

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. Convex sets A subset C \subseteq X of som ...

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 ge ...

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 (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...

geometry of numbers Geometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in \mathbb R^n, and the study of these lattices provides fundamental informatio ...

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 (or equivariant) transformati ...

, linear programming,

probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...

, game theory, etc.

Classification

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:Mathematics Subject Classification MSC2010, entry 52A "General convexity"
/ref> *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 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. Antimatroi ...

, which is one of the abstract models of convex sets.

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 (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...

and Archimedes, it became an independent branch of mathematics at the turn of the 20th century, mainly due to the works of

Hermann Brunn Karl Hermann Brunn (1 August 1862 – 20 September 1939) was a German mathematician, known for his work in convex geometry (see Brunn–Minkowski inequality) and in knot theory. Brunnian links are named after him, as his 1892 article "Über Ve ...

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. Bonnesen and W. Fenchel gave a comprehensive survey of convex geometry in

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.

Notes

References

Expository articles on convex geometry *K. Ball, ''An elementary introduction to modern convex geometry,'' in: Flavors of Geometry, pp. 1–58, Math. Sci. Res. Inst. Publ. Vol. 31, Cambridge Univ. Press, Cambridge, 1997, availabl
online
*M. Berger, ''Convexity,'' Amer. Math. Monthly, Vol. 97 (1990), 650—678. DOI
10.2307/2324573
*P. M. Gruber, ''Aspects of convexity and its applications,'' Exposition. Math., Vol. 2 (1984), 47—83. *V. Klee, ''What is a convex set?'' Amer. Math. Monthly, Vol. 78 (1971), 616—631, DOI
10.2307/2316569
Books on convex geometry *T. Bonnesen, W. Fenchel, ''Theorie der konvexen Körper,'' Julius Springer, Berlin, 1934. English translation: ''Theory of convex bodies,'' BCS Associates, Moscow, ID, 1987. *R. J. Gardner, ''Geometric tomography,'' Cambridge University Press, New York, 1995. Second edition: 2006. * P. M. Gruber, ''Convex and discrete geometry,'' Springer-Verlag, New York, 2007. *P. M. Gruber, J. M. Wills (editors), ''Handbook of convex geometry. Vol. A. B,'' North-Holland, Amsterdam, 1993. *G. Pisier, ''The volume of convex bodies and Banach space geometry,'' Cambridge University Press, Cambridge, 1989. *R. Schneider, ''Convex bodies: the Brunn-Minkowski theory,'' Cambridge University Press, Cambridge, 1993; Second edition: 2014. *A. C. Thompson, ''Minkowski geometry,'' Cambridge University Press, Cambridge, 1996. Articles on history of convex geometry *W. Fenchel, ''Convexity through the ages,'' (Danish) Danish Mathematical Society (1929—1973), pp. 103–116, Dansk. Mat. Forening, Copenhagen, 1973. English translation: ''Convexity through the ages,'' in: P. M. Gruber, J. M. Wills (editors), Convexity and its Applications, pp. 120–130, Birkhauser Verlag, Basel, 1983. *P. M. Gruber, ''Zur Geschichte der Konvexgeometrie und der Geometrie der Zahlen,'' in: G. Fischer, et al. (editors), Ein Jahrhundert Mathematik 1890—1990, pp. 421–455, Dokumente Gesch. Math., Vol. 6, F. Wieweg and Sohn, Braunschweig; Deutsche Mathematiker Vereinigung, Freiburg, 1990. *P. M. Gruber, ''History of convexity,'' in: P. M. Gruber, J. M. Wills (editors), Handbook of convex geometry. Vol. A, pp. 1–15, North-Holland, Amsterdam, 1993.

External links

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Classification

Historical note

See also

Notes

References

External links