Let M² 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 M², 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm142-2-3,
author = {Horst Martini and Anatoliy Shcherba},
title = {Upper estimates on self-perimeters of unit circles for gauges},
journal = {Colloquium Mathematicae},
volume = {144},
year = {2016},
pages = {179-210},
zbl = {1334.28008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm142-2-3}
}
Horst Martini; Anatoliy Shcherba. Upper estimates on self-perimeters of unit circles for gauges. Colloquium Mathematicae, Tome 144 (2016) pp. 179-210. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm142-2-3/