Abstract
A metric space (X, D) is called circular if it is isometric to a subspace of a metric circle, that is, a circle in which distances are measured by the length of the shorter arc connecting them. We show that the following three conditions are equivalent: (1) X is circular, (2) every 4-point subset of X can be labeled as a, b, c, d so that \(D(a,b)+D(b,c)=D(a,c)\), \(D(b,c)+D(c,d)=D(b,d)\) holds, and (3) every 4-point subset of X is circular.
This paper was supported in part by the National Natural Science Foundation of China under Grant No. 11471209.
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Notes
- 1.
As always, given any metric space \(X=(X,D)\) and any two points \(x,y\in X\), we denote by [x, y] the interval \([x,y]=[x,y]_{D}:=\{z\in X: D(x,y)=D(x,z)+D(z,y)\}\) spanned by x and y relative to D. Recall that \(x,y,z,u\in X, y\in [x,z]\), and \(z\in [x,u]\) always implies \(y\in [x,u]\) and \(z\in [y,u]\).
References
Apostol, T.M.: Ptolemy’s inequality and the chordal metric. Math. Mag. 40, 233–235 (1967)
Berger, M.: Geometry I. Springer, Heidelberg (1967). 1977. MAA
Blumenthal, L.: A new concept in distance geometry with applications to spherical subsets. Bull. Am. Math. Soc. 47(6), 435–443 (1941)
Blumenthal, L.: Theory and Applications of Distance Geometry, 2nd edn. Chelsce Pub. Co., Bronx (1970)
Chepoi, V., Fichet, B.: A note on circular decomposable metrics. Geom. Ded. 69, 237–240 (1998)
Deza, E., Deza, M.M.: Dictionary of Distances. Elsevier, Tokyo (2006)
Deza, M., Laurent, M.: Geometry of Cuts and Metrics. Algorithms and Combinatorics, vol. 15. Springer, Heidelberg (1997). https://doi.org/10.1007/978-3-642-04295-9
Robinson, P.L.: The sphere is not flat. Am. Math. Monthly 113, 171–172 (2006)
Tóth, L.F.: Langerungen in der Ebene auf Kugel unt im Raum. Springer, Heidelberg (1972). https://doi.org/10.1007/978-3-642-65234-9
Acknowledgment
We wish to thank Michel Deza for various interesting and helpful discussions of the topic of this paper.
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Dress, A.W.M., Maehara, H., Pang, S.X.M., Zeng, Z. (2019). On the Structure of Discrete Metric Spaces Isometric to Circles. In: Du, DZ., Li, L., Sun, X., Zhang, J. (eds) Algorithmic Aspects in Information and Management. AAIM 2019. Lecture Notes in Computer Science(), vol 11640. Springer, Cham. https://doi.org/10.1007/978-3-030-27195-4_8
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