The term convex geometry is also used in
combinatorics as an alternate name for an
antimatroid, which is one of the abstract models of convex sets.
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
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 spaceRn. 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.
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:
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.