Geometry
Projecting asphere to aplane
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Concepts Features Dimension Straightedge and compass constructions Angle Curve Diagonal Orthogonality (Perpendicular) Parallel Vertex Congruence Similarity Symmetry
Zero-dimensional Point
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Three-dimensional Surface area Volume Polyhedron Platonic Solid Tetrahedron cuboid Cube Octahedron Dodecahedron Icosahedron Pyramid Solid of revolution Sphere Great circle Cylinder Cone
Four-/other-dimensional 4-polytope Simplex 5-cell Hypercube Tesseract n-sphere Hypersphere
Geometers
by name Aida Aryabhata Ahmes Alhazen Apollonius Archimedes Atiyah Baudhayana Bolyai Brahmagupta Cartan Chern Coxeter Descartes Euclid Euler Gauss Gromov Hilbert Huygens Jyeṣṭhadeva Kātyāyana Khayyám Klein Lobachevsky Manava Minkowski Minggatu Pascal Pythagoras Parameshvara Poincaré Riemann Sakabe Sijzi al-Tusi Veblen Virasena Yang Hui al-Yasamin Zhang List of geometers
by period BCE Ahmes Baudhayana Manava Pythagoras Euclid Archimedes Apollonius 1–1400s Zhang Kātyāyana Aryabhata Brahmagupta Virasena Alhazen Sijzi Khayyám al-Yasamin al-Tusi Yang Hui Parameshvara 1400s–1700s Jyeṣṭhadeva Descartes Pascal Huygens Minggatu Euler Sakabe Aida 1700s–1900s Gauss Lobachevsky Bolyai Riemann Klein Poincaré Hilbert Minkowski Cartan Veblen Coxeter Chern Present day Atiyah Gromov
v t e

Inmathematics,convex geometry is the branch ofgeometry studyingconvex sets, mainly inEuclidean 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 theMathematics Subject Classification MSC2010,^[1] the mathematical disciplineConvex and Discrete Geometry includes three major branches:^[2]

• 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:^[3]

• 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 inn 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

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 ofEuclid andArchimedes, it became an independent branch of mathematics at the turn of the 20th century, mainly due to the works ofHermann Brunn andHermann Minkowski in dimensions two and three. A big part of their results was soon generalized to spaces of higher dimensions, and in 1934T. Bonnesen andW. Fenchel gave a comprehensive survey of convex geometry inEuclidean spaceRⁿ. Further development of convex geometry in the 20th century and its relations to numerous mathematical disciplines are summarized in theHandbook of convex geometry edited by P. M. Gruber and J. M. Wills.

• List of convexity topics

• Ball, K. (1997). "An elementary introduction to modern convex geometry".Flavors of Geometry(PDF). Math. Sci. Res. Inst. Publ. Vol. 31. Cambridge: Cambridge Univ. Press. pp. 1–58.
• Berger, M. (1990). "Convexity".Amer. Math. Monthly.97:650–678.doi:10.2307/2324573.
• Gruber, P. M. (1984). "Aspects of convexity and its applications".Exposition. Math.2:47–83.
• Klee, V. (1971). "What is a convex set?".Amer. Math. Monthly.78:616–631.doi:10.2307/2316569.

• Bonnesen, T.; Fenchel, W. (1987) [1934].Theorie der konvexen Körper [Theory of convex bodies]. Moscow, ID: BCS Associates.
• Gardner, R. J. (2006) [1995].Geometric tomography (2nd ed.). New York: Cambridge University Press.
• Gruber, P. M. (2007).Convex and discrete geometry. New York: Springer-Verlag.
• Gruber, P. M.; Wills, J. M., eds. (1993).Handbook of convex geometry. Vol. A. B. Amsterdam: North-Holland.
• Pisier, G. (1989).The volume of convex bodies and Banach space geometry. Cambridge: Cambridge University Press.
• Schneider, R. (2014) [1993].Convex bodies: the Brunn-Minkowski theory (2nd ed.). Cambridge: Cambridge University Press.
• Thompson, A. C. (1996).Minkowski geometry. Cambridge: Cambridge University Press.
• Balestro, Vitor; Martini, Horst; Teixeira, Ralph (2024).Convexity from the Geometric Point of View. Cornerstones. Cham: Springer International Publishing.doi:10.1007/978-3-031-50507-2.ISBN 978-3-031-50506-5.

• Fenchel, W. (1983) [1973]. "Convexity through the ages". In Gruber, P. M.; Wills, J. M. (eds.).Convexity and its Applications. Basel: Birkhauser Verlag. pp. 120–130.doi:10.1007/978-3-0348-5858-8_6.
• Gruber, Peter Manfred (1990), Fischer, Gerd; Hirzebruch, Friedrich; Scharlau, Winfried; Törnig, Willi (eds.),"Zur Geschichte der Konvexgeometrie und der Geometrie der Zahlen",Ein Jahrhundert Mathematik 1890–1990: Festschrift zum Jubiläum der DMV (in German), Wiesbaden: Vieweg+Teubner Verlag, pp. 421–455,doi:10.1007/978-3-322-80265-1_9,ISBN 978-3-322-80265-1
• Gruber, P. M. (1993). "History of convexity". In Gruber, P. M.; Wills, J. M. (eds.).Handbook of convex geometry. Vol. A. Amsterdam: North-Holland. pp. 1–15.

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