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 onlyif $D$ is linearly equivalent to a regular hexagon. The upper boundis attained if and only if $D$ is a parallelogram.

Note that 1/2 the perimeter of the unit disc is a constantbetween 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 aMinkowski 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.

单词	GolabsTheorem
释义	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 onlyif $D$ is linearly equivalent to a regular hexagon. The upper boundis attained if and only if $D$ is a parallelogram. Note that 1/2 the perimeter of the unit disc is a constantbetween 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 aMinkowski 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.
随便看	SardsTheorem1 Sard’s theorem Sard’s theorem SatisfactionRelation satisfaction relation Saturate saturate Saturated saturated Saturatedset saturated (set) Scalar scalar ScalarFactorTransferRules scalar factor transfer rules ScalarMap scalar map ScaleneTriangle scalene triangle ScalingOfTheOpenBallInANormedVectorSpace scaling of the open ball in a normed vector space ScatteredSpace scattered space Scenario scenario