Golab’s theorem

Theorem. Let D be the unit disc of a Minkowski plane and let (D) denote the Minkowski length (http://planetmath.org/LengthOfCurveInAMetricSpace) of the boundary of D. Then 6(D)8. The lower bound is attained if and only if D is linearly equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to a regular hexagon. The upper bound is attained if and only if D is a parallelogramMathworldPlanetmath.

Note that 1/2 the perimeterPlanetmathPlanetmath of the unit disc is a constant between 3 and 4. The special case of the 2-norm yields a constant, which is known as π. So Golab’s theorem is that ”pi” for a Minkowski plane is always between 3 and 4.


  • GO S. Golab, Quelques problèmes métriques de la géometrie de Minkowski, Trav. l’Acad. Mines Cracovie 6 (1932) 1-79.
  • PE C.M. Petty, GeometryMathworldPlanetmath of the Minkowski plane, Riv. Mat. Univ. Parma (4) 𝟞 (1955) 269-292.
  • SC J.J. Schäefer, Inner diameterMathworldPlanetmath, perimeter, and girth of spheres, Math. Ann. 𝟙𝟟𝟛 (1967) 59-79.
  • ACT A.C. Thompson, Minkowski Geometry, Encyclopedia of Mathematics and its Applications, 63, Cambridge University Press, Cambridge, 1996.
Title Golab’s theorem
Canonical name GolabsTheorem
Date of creation 2013-03-22 16:50:33
Last modified on 2013-03-22 16:50:33
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 7
Author Mathprof (13753)
Entry type Theorem
Classification msc 46B20