Golab’s theorem
Theorem. Let be the unit disc of a Minkowski plane and let
denote the Minkowski length (http://planetmath.org/LengthOfCurveInAMetricSpace) of the boundary of . Then
. The lower bound is attained if and only
if is linearly equivalent to a regular hexagon. The upper bound
is attained if and only if is a parallelogram
.
Note that 1/2 the perimeter 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 . So Golab’s theorem is that ”pi” for a
Minkowski plane is always between 3 and 4.
References
- GO S. Golab, Quelques problèmes métriques de la géometrie de Minkowski, Trav. l’Acad. Mines Cracovie (1932) 1-79.
-
PE
C.M. Petty, Geometry
of the Minkowski plane, Riv. Mat. Univ. Parma (4) (1955) 269-292.
-
SC
J.J. Schäefer, Inner diameter
, perimeter, and girth of spheres, Math. Ann. (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 |