Upper estimates on self-perimeters of unit circles for gauges

Tom 142 / 2016

Horst Martini, Anatoliy Shcherba Colloquium Mathematicum 142 (2016), 179-210 MSC: 28A75, 46B20, 52A10, 52A21, 52A38, 52A40. DOI: 10.4064/cm142-2-3


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.


  • Horst MartiniFaculty of Mathematics
    Technical University of Chemnitz
    09107 Chemnitz, Germany
  • Anatoliy ShcherbaDepartment of Industrial Computer Technologies
    Cherkasy State Technological University
    Shevchenko Blvd., 460
    Cherkasy, 18006, Ukraine

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