convex geometry
{{Short description|Branch of geometry}}
{{about|the study of convex sets|the combinatorial set systems|antimatroid}}
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.
Classification
According to the Mathematics Subject Classification MSC2010,[http://www.msc2010.org/mscwiki/index.php?title=MSC2010 Website of Mathematics Subject Classification MSC2010] the mathematical discipline Convex and Discrete Geometry includes three major branches:[http://www.msc2010.org/mscwiki/index.php?title=52-XX Mathematics Subject Classification MSC2010, entry 52 "Convex and discrete geometry"]
- 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:[http://www.msc2010.org/mscwiki/index.php?title=52Axx Mathematics Subject Classification MSC2010, entry 52A "General convexity"]
- 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
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 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. Bonnesen and W. Fenchel gave a comprehensive survey of convex geometry in Euclidean space 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.
See also
Notes
{{reflist}}
References
= Expository articles on convex geometry =
- {{cite book
| last = Ball
| first = K.
| chapter = An elementary introduction to modern convex geometry
| editor-last =
| editor-first =
| title = Flavors of Geometry
| series = Math. Sci. Res. Inst. Publ.
| volume = 31
| publisher = Cambridge Univ. Press
| location = Cambridge
| date = 1997
| pages = 1–58
| url = https://www.math.uchicago.edu/~shmuel/AAT-readings/Combinatorial%20Geometry,%20Concentration,%20Real%20Algebraic%20Geometry/ball.pdf
}}
- {{cite journal
| last = Berger
| first = M.
| title = Convexity
| journal = Amer. Math. Monthly
| volume = 97
| date = 1990
| pages = 650–678
| doi = 10.2307/2324573
}}
- {{cite journal
| last = Gruber
| first = P. M.
| title = Aspects of convexity and its applications
| journal = Exposition. Math.
| volume = 2
| date = 1984
| pages = 47–83
}}
- {{cite journal
| last = Klee
| first = V.
| title = What is a convex set?
| journal = Amer. Math. Monthly
| volume = 78
| date = 1971
| pages = 616–631
| doi = 10.2307/2316569
}}
= Books on convex geometry =
- {{cite book
| last1 = Bonnesen
| first1 = T.
| last2 = Fenchel
| first2 = W.
| title = Theorie der konvexen Körper
| publisher = BCS Associates
| location = Moscow, ID
| date = 1987
| orig-year = 1934
|trans-title=Theory of convex bodies}}
- {{cite book
| last = Gardner
| first = R. J.
| title = Geometric tomography
| edition = 2nd
| publisher = Cambridge University Press
| location = New York
| date = 2006
| orig-year = 1995
}}
- {{cite book
| last = Gruber
| first = P. M.
| author-link = Peter M. Gruber
| title = Convex and discrete geometry
| publisher = Springer-Verlag
| location = New York
| date = 2007
}}
- {{cite book
| editor-last1 = Gruber
| editor-first1 = P. M.
| editor-last2 = Wills
| editor-first2 = J. M.
| title = Handbook of convex geometry. Vol. A. B
| publisher = North-Holland
| location = Amsterdam
| date = 1993
}}
- {{cite book
| last = Pisier
| first = G.
| title = The volume of convex bodies and Banach space geometry
| publisher = Cambridge University Press
| location = Cambridge
| date = 1989
}}
- {{cite book
| last = Schneider
| first = R.
| title = Convex bodies: the Brunn-Minkowski theory
| edition = 2nd
| publisher = Cambridge University Press
| location = Cambridge
| date = 2014
| orig-year = 1993
}}
- {{cite book
| last = Thompson
| first = A. C.
| title = Minkowski geometry
| publisher = Cambridge University Press
| location = Cambridge
| date = 1996
}}
- {{Cite book |last=Balestro |first=Vitor |url=https://link.springer.com/10.1007/978-3-031-50507-2 |title=Convexity from the Geometric Point of View |last2=Martini |first2=Horst |last3=Teixeira |first3=Ralph |date=2024 |publisher=Springer International Publishing |isbn=978-3-031-50506-5 |series=Cornerstones |location=Cham |language=en |doi=10.1007/978-3-031-50507-2}}
= Articles on history of convex geometry =
- {{cite book
| last = Fenchel
| first = W.
| chapter = Convexity through the ages
| editor-last1 = Gruber
| editor-first1 = P. M.
| editor-last2 = Wills
| editor-first2 = J. M.
| title = Convexity and its Applications
| publisher = Birkhauser Verlag
| location = Basel
| date = 1983
| pages = 120–130
| orig-date = 1973
|url=https://link.springer.com/chapter/10.1007/978-3-0348-5858-8_6|doi=10.1007/978-3-0348-5858-8_6}}
- {{Citation |last=Gruber |first=Peter Manfred |title=Zur Geschichte der Konvexgeometrie und der Geometrie der Zahlen |date=1990 |work=Ein Jahrhundert Mathematik 1890–1990: Festschrift zum Jubiläum der DMV |pages=421–455 |editor-last=Fischer |editor-first=Gerd |url=https://doi.org/10.1007/978-3-322-80265-1_9 |place=Wiesbaden |publisher=Vieweg+Teubner Verlag |language=de |doi=10.1007/978-3-322-80265-1_9 |isbn=978-3-322-80265-1 |editor2-last=Hirzebruch |editor2-first=Friedrich |editor3-last=Scharlau |editor3-first=Winfried |editor4-last=Törnig |editor4-first=Willi}}
- {{cite book
| last = Gruber
| first = P. M.
| chapter = History of convexity
| editor-last1 = Gruber
| editor-first1 = P. M.
| editor-last2 = Wills
| editor-first2 = J. M.
| title = Handbook of convex geometry. Vol. A
| publisher = North-Holland
| location = Amsterdam
| date = 1993
| pages = 1–15
}}