Isometry classes of planes in $(\mathbb{R}^3,d_{\infty})$

Yıl 2019, Cilt: 48 Sayı: 4, 1110 - 1120, 08.08.2019

Mehmet Kılıç

Öz

We determine geodesics in $\mathbb{R}_{\infty}^n$ (i.e. $(\mathbb{R}^n,d_{\infty})$) and by using this, classify planes up to isometry in $\mathbb{R}_{\infty}^3$.

Anahtar Kelimeler

geodesic, sector, plane, isometry

Kaynakça

• [1] N. Aronszajn and P. Panitchpakdi, Extension of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6, 405–439, 1956.
• [2] M.R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, 1999.
• [3] D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics American Mathematical Society, USA, 2001.
• [4] M. Kılıç and S. Koçak, Tight Span of Subsets of The Plane With The Maximum Metric, Adv. Math. 301, 693–710, 2016.
• [5] A. Moezzi, The Injective Hull of Hyperbolic Groups, Dissertation ETH Zurich, No: 18860, 2010.
• [6] L. Nachbin, A Theorem of The Hahn-Banach Type For Linear Transformations, Trans. Amer. Math. Soc. 68, 28–46, 1950.
• [7] A. Papadopoulos, Metric Spaces, Convexity and Nonpositive Curvature, IRMA Lect. Math. Theor. Phys. European Mathematical Society, Germany, 2005.
• [8] M. Pavon, Injective Convex Polyhedra, Discrete Comput. Geom. doi:10.1007/s00454- 016-9810-6, 2016.