Convex geometry

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,^[1] the mathematical discipline Convex 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 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.

Notes

References

Expository articles on convex geometry

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.

Books on convex geometry

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.

Articles on history of convex geometry

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.