Estimating from Above the Perimeter of an Asymmetric Unit Disk in the Minkowski Plane
It is proved that each convex planar figure K ⊂ ℝ2 contains a point O such that the perimeter of K computed with respect to the Minkowski distance function of the pair (K, O) is at most 9. If K is a triangle, this estimate is sharp. Bibliography: 7 titles.
KeywordsDistance Function Unit Disk Asymmetric Unit Minkowski Plane Minkowski Distance
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- 1.H. Minkowski, Geometrie der Zahlen, Leipzig-Berlin (1910).Google Scholar
- 2.A. I. Shcherba, “On the estimate of the perimeter of the normalizing figure on the Minkowski plane,” in: 5th International Conference on Geometry and Topology in Memory of A. V. Pogorelov, Abstracts of Reports, Cherkassy (2003).Google Scholar
- 3.A. I. Shcherba, “On the estimate of the perimeter of the unit disk on the Minkowski plane,” in: Proc. Rubtsovsk Industry Institute, 12 (2003), pp. 96–107.Google Scholar
- 4.Yu. G. Reshetnyak, “An extremal problem in the theory of convex curves,” Uspekhi Mat. Nauk, 8, 125–126 (1953).Google Scholar
- 6.A. Besicovitch, “Measure of asymmetry of convex curves,” J. London Math. Soc., 23, 237–240 (1945).Google Scholar
- 7.V. V. Makeev, “On approximation of plane sections of a convex body,” Zap. Nauchn. Semin. POMI, 246, 174–183 (1997).Google Scholar