The maximal generalised roundness of finite metric spaces

Robertson, Gavin

doi:10.1090/proc/15298

The maximal generalised roundness of finite metric spaces

Author: Gavin Robertson
Journal: Proc. Amer. Math. Soc. 149 (2021), 407-411
MSC (2020): Primary 52C99; Secondary 43A35
DOI: https://doi.org/10.1090/proc/15298
Published electronically: October 20, 2020
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Abstract: A useful tool for computing the maximal generalised roundness of a finite metric space is a formula of Sánchez. Using well-known properties of the classical Cayley-Menger and Gramian matrices, we provide two simplifications of Sánchez's formula that in particular do not require the calculation of any matrix inverses. To demonstrate the usefulness of these two new formulas we provide a simpler proof of Murugan's classification of the subsets of the Hamming cube that have maximal generalised roundness greater than .

[1] I. Doust, G. Robertson, A. Stoneham, and A. Weston, Distance matrices of subsets of the Hamming cube, (Preprint).
[2] Per Enflo, On a problem of Smirnov, Ark. Mat. 8 (1969), 107–109. MR 415576, https://doi.org/10.1007/BF02589550
[3] Timothy Faver, Katelynn Kochalski, Mathav Kishore Murugan, Heidi Verheggen, Elizabeth Wesson, and Anthony Weston, Roundness properties of ultrametric spaces, Glasg. Math. J. 56 (2014), no. 3, 519–535. MR 3250263, https://doi.org/10.1017/S0017089513000438
[4] F. R. Gantmacher, `The theory of matrices.' Vol. I, Chelsea, New York, (1959), x + 374 pp.
[5] Poul Hjorth, Petr Lisonĕk, Steen Markvorsen, and Carsten Thomassen, Finite metric spaces of strictly negative type, Linear Algebra Appl. 270 (1998), 255–273. MR 1484084
[6] Jean-François Lafont and Stratos Prassidis, Roundness properties of groups, Geom. Dedicata 117 (2006), 137–160. MR 2231164, https://doi.org/10.1007/s10711-005-9019-y
[7] C. J. Lennard, A. M. Tonge, and A. Weston, Generalized roundness and negative type, Michigan Math. J. 44 (1997), no. 1, 37–45. MR 1439667, https://doi.org/10.1307/mmj/1029005619
[8] Hiroshi Maehara, Euclidean embeddings of finite metric spaces, Discrete Math. 313 (2013), no. 23, 2848–2856. MR 3106458, https://doi.org/10.1016/j.disc.2013.08.029
[9] Mathav Kishore Murugan, Supremal 𝑝-negative type of vertex transitive graphs, J. Math. Anal. Appl. 391 (2012), no. 2, 376–381. MR 2903138, https://doi.org/10.1016/j.jmaa.2012.02.042
[10] Stephen Sánchez, On the supremal 𝑝-negative type of finite metric spaces, J. Math. Anal. Appl. 389 (2012), no. 1, 98–107. MR 2876484, https://doi.org/10.1016/j.jmaa.2011.11.043
[11] I. J. Schoenberg, On certain metric spaces arising from Euclidean spaces by a change of metric and their imbedding in Hilbert space, Ann. of Math. (2) 38 (1937), no. 4, 787–793. MR 1503370, https://doi.org/10.2307/1968835
[12] I. J. Schoenberg, Metric spaces and positive definite functions, Trans. Amer. Math. Soc. 44 (1938), no. 3, 522–536. MR 1501980, https://doi.org/10.1090/S0002-9947-1938-1501980-0