Convex geometry

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In mathematics, convex geometry is a branch of geometry that studies the convex sets, especially in the Euclidean space. Convex sets occur naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, number geometry, integral geometry, linear programming, probability theory, etc.

Video Convex geometry

Classification

According to Subject Mathematics Classification of MSC2010, the mathematical disciplines Convex and Discrete Geometry include three main branches:

• general convexity
• polytopes and polyhedra
• discrete geometry

The common convex is further subdivided as follows:

• axiomatic and general lies
• convex without dimension limit
• convex in vector topology space
• convex in 2 dimensions (including convex curves)
• convex in 3 dimensions (including convex surface)
• set convex within n dimensions (including convex hypersurfaces)
• Banach space of limited dimension
• set of random convex and integral geometry
• asymptotic theory of convex body
• approximation by convex set
• convex variants (star-shaped, ( m, n ) - convex, etc.)
• Helly-type theorems and geometric transversal theory
• another matter of combinatorial convexity
• length, breadth, volume
• mixed volume and related topics
• inequalities and extreme problems
• convex function and convex program
• hooked and hyperbolic

The term convex geometry is also used in combinatorics as the name for one of the abstract models of convex sets, which is equivalent to antimatroid.

Maps Convex geometry

Historical records

Convex geometry is a relatively young mathematical discipline. Although the first known contribution to convex geometry dates back to ancient times 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, largely because of the works of Hermann Brunn and Hermann Minkowski in dimensions two and three. Most of their results were immediately generalizable to the higher dimension spaces, and in 1934 T. Bonnesen and W. Fenchel gave a comprehensive survey of convex geometry in the Euclidean space R ⁿ . The further development of convex geometry in the 20th century and its relation to various mathematical disciplines is summarized in the Convex geometry handbook edited by P. M. Gruber and J. M. Wills.

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See also

• List of convexity topics

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Note

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References

Expository articles on convex geometry

• K. Ball, Basic introduction to modern convex geometry, in: Flavors of Geometry, pp. 1-58, Math. Sci. Res. Inst. Publ. Vol. 31, Cambridge Univ. Press, Cambridge, 1997, available online.
• M. Berger, Convexity, Amer. Mathematics. Monthly, Vol. 97 (1990), 650--678. DOI: 10.2307/2324573
• P. M. Gruber, Aspects of convexity and its application, Exposition. Mathematics., Vol. 2 (1984), 47--83.
• V. Klee, What is convex? Amer. Mathematics. Monthly, Vol. 78 (1971), 616--631, DOI: 10.2307/2316569

Book on convex geometry

• T. Bonnesen, W. Fenchel, Theorie der konvexen KÃÆ'¶rper, Julius Springer, Berlin, 1934. English translation: Convex body theory, 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 (editor), Handbook of convex geometry. Vol. A. B, North Holland, Amsterdam, 1993.
• G. Pisier, Convex body volume and Banach space geometry, Cambridge University Press, Cambridge, 1989.
• R. Schneider, Convex body: Brunn-Minkowski's theory, Cambridge University Press, Cambridge, 1993.
• A. C. Thompson, Minkowski geometry, Cambridge University Press, Cambridge, 1996.
• A. Koldobsky, V. Yaskin, Interface between Convex Geometry and Harmonic Analysis, American Mathematical Society, Providence, Rhode Island, 2008.

Article tentang riwayat geometri cembung

• W. Fenchel, Convexity through the ae, (Danish) Danish Mathematical Society (1929-1973), hlm. 103-116, Dansk. Tikar. Forening, Copenhagen, 1973. Terjemahan bahasa Inggris: Convexity sepanjang masa, dalam: PM Gruber, JM Wills (editor), Convexity dan Aplikasi-nya, pp.Ã, 120-130, Birkhauser Verlag, Basel , 1983.
• P. M. Gruber, On the History of Convex Geometry and the Geometry of Numbers, di: G. Fischer, dkk. (editor), A Century of Mathematics 1890--1990, pp.à, 421-455, Documents Gesch. Matematika., Vol. 6, F. Wieweg dan Sohn, Braunschweig; German Mathematician Association, Freiburg, 1990.
• P.M. Gruber, Sejarah konveksitas, dalam: P.M. Gruber, J.M. Wills (editor), Buku Pegangan geometri cembung. Vol. A, pp.Ã, 1-15, North-Holland, Amsterdam, 1993.

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