Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access December 5, 2016

Constant Distortion Embeddings of Symmetric Diversities

  • David Bryant and Paul F. Tupper


Diversities are like metric spaces, except that every finite subset, instead of just every pair of points, is assigned a value. Just as there is a theory of minimal distortion embeddings of fiite metric spaces into L1, there is a similar, yet undeveloped, theory for embedding finite diversities into the diversity analogue of L1 spaces. In the metric case, it iswell known that an n-point metric space can be embedded into L1 withO(log n) distortion. For diversities, the optimal distortion is unknown. Here, we establish the surprising result that symmetric diversities, those in which the diversity (value) assigned to a set depends only on its cardinality, can be embedded in L1 with constant distortion.


[1] N. Alon and P. Pudlák. Equilateral sets in `n p. Geometric & Functional Analysis GAFA, 13(3):467–482, 2003. 10.1007/s00039-003-0418-7Search in Google Scholar

[2] D. Bryant and S. Klaere. The link between segregation and phylogenetic diversity. J. Math. Biol., 64(1-2):149–162, 2012. 10.1007/s00285-011-0409-5Search in Google Scholar PubMed

[3] D. Bryant and P. F. Tupper. Hyperconvexity and tight-span theory for diversities. Advances in Mathematics, 231(6):3172– 3198, 2012. 10.1016/j.aim.2012.08.008Search in Google Scholar

[4] D. Bryant and P. F. Tupper. Diversities and the geometry of hypergraphs. Discrete Math. Theor. Comput. Sci., 16(2):1–20, 2014. 10.46298/dmtcs.2080Search in Google Scholar

[5] D. Bryant, A. Nies, and P. Tupper. A diversity analogue of the Urysohn metric space. arXiv preprint arXiv:1509.07173, 2015. Search in Google Scholar

[6] R. Espínola and A. Fernández-León. Fixed point theory in hyperconvex metric spaces. In Topics in Fixed Point Theory, pages 101–158. Springer, 2014. 10.1007/978-3-319-01586-6_4Search in Google Scholar

[7] R. Espínola and B. Piaµek. Diversities, hyperconvexity and fixed points. Nonlinear Analysis: Theory, Methods & Applications, 95:229 – 245, 2014. 10.1016/ in Google Scholar

[8] W. Kirk and N. Shahzad. Diversities. In Fixed Point Theory in Distance Spaces, pages 153–158. Springer, 2014. 10.1007/978-3-319-10927-5_15Search in Google Scholar

[9] P. Kumar, N. Komodakis, and N. Paragios. (Hyper)-Graphs Inference via Convex Relaxations and Move Making Algorithms: Contributions and Applications in artificial vision. Technical report, Inria, 2015. 10.1561/9781680831399Search in Google Scholar

[10] N. Linial, E. London, and Y. Rabinovich. The geometry of graphs and some of its algorithmic applications. Combinatorica, 15(2):215–245, 1995. 10.1007/BF01200757Search in Google Scholar

[11] J. Matoušek. Thirty-three miniatures: Mathematical and Algorithmic applications of Linear Algebra, volume 53. American Mathematical Soc., 2010. 10.1090/stml/053Search in Google Scholar

[12] B. Piaµek. On the gluing of hyperconvex metrics and diversities. Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica, 13(1):65–76, 2014. 10.2478/aupcsm-2014-0006Search in Google Scholar

[13] A. Poelstra. On the topological and uniform structure of diversities. Journal of Function Spaces and Applications, 2013. 10.1155/2013/675057Search in Google Scholar

[14] M. Steel. Tracing evolutionary links between species. The American Mathematical Monthly, 121(9):771–792, 2014. 10.4169/amer.math.monthly.121.09.771Search in Google Scholar

[15] M. Taylor. Measure theory and integration, volume 76 of Graduate Studies inMathematics. AmericanMathematical Society, 2006. Search in Google Scholar

Received: 2016-5-11
Accepted: 2016-11-8
Published Online: 2016-12-5

© 2016 David Bryant and Paul F. Tupper

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 6.6.2023 from
Scroll to top button