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A flat strip theorem for Ptolemaic spaces


Miao, Relong; Schroeder, Viktor (2013). A flat strip theorem for Ptolemaic spaces. Mathematische Zeitschrift, 274(1-2):461-470.

Abstract

A metric space (X,d) is called Ptolemaic space (PT space) if the inequality |xy||zw|≤|xz||yw|+|xw||yz| holds for each quadruple of points x, y, z and w in X. Here |xy|:=d(x,y) denotes the distance of two points x and y. The main result proven in the paper is the following: If (X,d) is a PT space which is also a geodesic space (i.e., for each pair x,y there exists a geodesic connecting x and y) and if X is moreover homeomorphic to ℝ×[0,1] then X is isometric to a flat strip ℝ×[0,a]⊂ℝ 2 with its Euclidean metric. The authors also give a new short proof of the fact that a proper geodesic PT space is always strictly distance convex. (See also Th. Foertsch and V. Schroeder [Trans. Am. Math. Soc. 363, No. 6, 2891–2906 (2011; Zbl 1220.53092)]). This also indicates a positive answer to the open question if every proper geodesic PT space is also a CAT(0)-space.

Abstract

A metric space (X,d) is called Ptolemaic space (PT space) if the inequality |xy||zw|≤|xz||yw|+|xw||yz| holds for each quadruple of points x, y, z and w in X. Here |xy|:=d(x,y) denotes the distance of two points x and y. The main result proven in the paper is the following: If (X,d) is a PT space which is also a geodesic space (i.e., for each pair x,y there exists a geodesic connecting x and y) and if X is moreover homeomorphic to ℝ×[0,1] then X is isometric to a flat strip ℝ×[0,a]⊂ℝ 2 with its Euclidean metric. The authors also give a new short proof of the fact that a proper geodesic PT space is always strictly distance convex. (See also Th. Foertsch and V. Schroeder [Trans. Am. Math. Soc. 363, No. 6, 2891–2906 (2011; Zbl 1220.53092)]). This also indicates a positive answer to the open question if every proper geodesic PT space is also a CAT(0)-space.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2013
Deposited On:14 Nov 2013 10:41
Last Modified:05 Apr 2016 17:08
Publisher:Springer
ISSN:0025-5874
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.1007/s00209-012-1078-9
Related URLs:http://link.springer.com/article/10.1007/s00209-012-1078-9 (Publisher)

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