Golab’s theorem

Theorem. Let $D$ be the unit disc of a Minkowski plane and let $\ell(\partial D)$ denote the Minkowski length (http://planetmath.org/LengthOfCurveInAMetricSpace) of the boundary of $D$ . Then $6\leq\ell(\partial D)\leq 8$ . The lower bound is attained if and only if $D$ is linearly equivalent to a regular hexagon. The upper bound is attained if and only if $D$ 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 $\pi$ . 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 $\mathbb{6}$ (1932) 1-79.
PE C.M. Petty, Geometry of the Minkowski plane, Riv. Mat. Univ. Parma (4) $\mathbb{6}$ (1955) 269-292.
SC J.J. Schäefer, Inner diameter, perimeter, and girth of spheres, Math. Ann. $\mathbb{173}$ (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