Upper estimates on self-perimeters of unit circles for gauges
Volume 142 / 2016
Colloquium Mathematicum 142 (2016), 179-210
MSC: 28A75, 46B20, 52A10, 52A21, 52A38, 52A40.
DOI: 10.4064/cm142-2-3
Abstract
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.
