Golab’s theorem
Theorem. Let $D$ be the unit disc of a Minkowski plane and let $\mathrm{\ell}(\partial D)$ denote the Minkowski length (http://planetmath.org/LengthOfCurveInAMetricSpace) of the boundary of $D$. Then $6\le \mathrm{\ell}(\partial D)\le 8$. 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 $\mathrm{6}$ (1932) 1-79.
- PE C.M. Petty, Geometry^{} of the Minkowski plane, Riv. Mat. Univ. Parma (4) $\mathrm{\U0001d7de}$ (1955) 269-292.
- SC J.J. Schäefer, Inner diameter^{}, perimeter, and girth of spheres, Math. Ann. $\mathrm{\U0001d7d9\U0001d7df\U0001d7db}$ (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 |