Hostname: page-component-cb9f654ff-mnl9s Total loading time: 0 Render date: 2025-08-18T09:41:05.287Z Has data issue: false hasContentIssue false
  • English
  • Français

Isoperimetric problems for polytopes with a given number of vertices

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Károly Böröczky  and
Károly Böröczky Jr.
Károly Böröczky
Affiliation:
Math. Inst. Hung. Acad. Sci., Budapest, Pf. 127, 1364 Hungary.
Károly Böröczky Jr.
Affiliation:
Math. Inst. Hung. Acad. Sci., Budapest, Pf. 127, 1364 Hungary.
Get access

Extract

The prototype of isoperimetric problems is to minimize the surface area of a convex body with given volume. The minimal body is naturally the suitable ball. The solution to this problem in the planar case was already known to the ancient Greeks. In the higher dimensional cases, the first proofs were provided with the help of Steiner's symmetrization method towards the end of the last century. Important later contributors are, among others, Minkowski, Blaschke, Hadwiger. By their work, the optimality of the ball has been also verified for a much wider class of sets (see [14]).

MSC classification

Secondary: 52A40: Inequalities and extremum problems

Information

Type
Research Article
Information
Mathematika , Volume 43 , Issue 2 , December 1996 , pp. 237 - 254
Copyright
Copyright © University College London 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1.Bonnesen, T. and Fenchel, W.. Theorie der konvexen Körper (Springer, Berlin, 1934).Google Scholar
2.Böröczky, K. Jr Some Extremal Properties of the Regular Simplex. In Proceedings of the Conference on Intuitive Geometry, Szeged, 1994.Google Scholar
3.Capoyleas, V. and Pach, J.. On the perimeter of a point set in the plane. DIMACS Ser. in Disc. Math, and Theo. Camp. Sci., 6 (1991), 67 76.CrossRefGoogle Scholar
4.Cheeger, J.Müller, W. and Schrader, R.. On the curvature of piecewise flat spaces. Comm. Math. Phys., 92 (1984), 405 454.CrossRefGoogle Scholar
5.Chisini, O.. Sulla teoria elementare degli isoperimetri. In Enriques, F., editor, Questioni Riguardanti le Mathematiche Elementari, 310 (1927), 201 310.Google Scholar
6.Connelly, R.. Rigidity. In Gruber, P. M. and Wills, J. M., editors, Handbook of Convex Geometry (1993), 223 271.CrossRefGoogle Scholar
7.Fejes Tóth, L.. Lagerungen in der Ebene. aufder Kugel und im Raum (Springer-Verlag, Berlin, 1972).CrossRefGoogle Scholar
8.Florian, A.. Extremum problems for convex discs and polyhedra. In Gruber, P. M. and Wills, J. M., editors, Handbook of Convex Geometry (1993), 177 221.CrossRefGoogle Scholar
9.Gritzmann, P.Wills, J. M. and Wrase, D.. A new isoperimetric inequality. J. Angew. Math., 379 (1987), 22–30.Google Scholar
10.Groemer, H.. Stability of Geometric Inequalities. In Handbook of Convexity (1993), 124 150.CrossRefGoogle Scholar
11.Hadwiger, H.. Vorlesung über Inhalt, Oberfläche und Isometric (Springer-Verlag, Berlin, 1957).CrossRefGoogle Scholar
12.Kneser, H.. Der Simplexinhalt in der nichteuklidischen Geometrie. Detsch. Math., 1 (1936), 337–340.Google Scholar
13.McMullen, P.. Valuations and Dissections. In Gruber, P. M. and Wills, J. M., editors. Handbook of Convex Geometry (1993), 933 988.CrossRefGoogle Scholar
14.Talenti, G.. The standard isoperimetric theorem. In Gruber, P. M. and Wills, J. M., editors, Handbook of Convex Geometry (1993), 73 123.CrossRefGoogle Scholar