Upper estimates on self-perimeters of unit circles for gauges

Horst Martini ¹   ; Anatoliy Shcherba ²

¹ Faculty of Mathematics Technical University of Chemnitz 09107 Chemnitz, Germany
² Department of Industrial Computer Technologies Cherkasy State Technological University Shevchenko Blvd., 460 Cherkasy, 18006, Ukraine

Colloquium Mathematicum, Tome 142 (2016) no. 2, pp. 179-210

Résumé

Let $M^2$ denote a Minkowski plane, i.e., an affine plane whose metric is a gauge induced by a compact convex figure $B$ which, as a unit circle of $M^2$, is not necessarily centered at the origin. Hence the self-perimeter of $B$ has two values depending on the orientation of measuring it. We prove that this self-perimeter of $B$ is bounded from above by the four-fold self-diameter of $B$. In addition, we derive a related non-trivial result on Minkowski planes whose unit circles are quadrangles.

Détail

DOI : 10.4064/cm142-2-3

Keywords: denote minkowski plane affine plane whose metric nbsp gauge induced compact convex figure which unit circle necessarily centered origin hence self perimeter has values depending orientation measuring prove self perimeter bounded above four fold self diameter addition derive related non trivial result minkowski planes whose unit circles quadrangles